P K Nag Exercise problems - Solved
Thermodynamics
Contents
Chapter-1: Introduction
Chapter-2: Temperature
Chapter-3: Work and Heat Transfer
Chapter-4: First Law of Thermodynamics
Chapter-5: First Law Applied to Flow Process
Chapter-6: Second Law of Thermodynamics
Chapter-7: Entropy
Chapter-8: Availability & Irreversibility
Chapter-9: Properties of Pure Substances
Chapter-10: Properties of Gases and Gas Mixture
Chapter-11: Thermodynamic Relations
Chapter-12: Vapour Power Cycles
Chapter-13: Gas Power Cycles
Chapter-14: Refrigeration Cycles
Page 2 of 265
Introduction
Chapter 1
1.
Introduction
Some Important Notes
ƒ
ƒ
ƒ
Microscopic thermodynamics or statistical thermodynamics
Macroscopic thermodynamics or classical thermodynamics
A quasi-static process is also called a reversible process
Intensive and Extensive Properties
Intensive property: Whose value is independent of the size or extent i.e. mass of the system.
e.g., pressure p and temperature T.
Extensive property: Whose value depends on the size or extent i.e. mass of the system (upper
case letters as the symbols). e.g., Volume, Mass (V, M). If mass is increased, the value of
extensive property also increases. e.g., volume V, internal energy U, enthalpy H, entropy S, etc.
Specific property: It is a special case of an intensive property. It is the value of an extensive
property per unit mass of system. (Lower case letters as symbols) e.g: specific volume, density
(v, ρ).
Concept of Continuum
The concept of continuum is a kind of idealization of the continuous description of matter where
the properties of the matter are considered as continuous functions of space variables. Although
any matter is composed of several molecules, the concept of continuum assumes a continuous
distribution of mass within the matter or system with no empty space, instead of the actual
conglomeration of separate molecules.
Describing a fluid flow quantitatively makes it necessary to assume that flow variables
(pressure, velocity etc.) and fluid properties vary continuously from one point to another.
Mathematical descriptions of flow on this basis have proved to be reliable and treatment of fluid
medium as a continuum has firmly become established.
For example density at a point is normally defined as
⎛ m⎞
ρ = lim ⎜
+∀→0 +∀ ⎟
⎝
⎠
Here +∀ is the volume of the fluid element and m is the mass
If +∀ is very large ρ is affected by the in-homogeneities in the fluid medium. Considering
another extreme if +∀ is very small, random movement of atoms (or molecules) would change
their number at different times. In the continuum approximation point density is defined at the
smallest magnitude of +∀ , before statistical fluctuations become significant. This is called
continuum limit and is denoted by +∀C .
⎛ m ⎞
ρ = lim ⎜
⎟
+∀→+∀
⎝ +∀ ⎠
C
Page 3 of 265
Introduction
Chapter 1
One of the factors considered important in determining the validity of continuum model is
molecular density. It is the distance between the molecules which is characterized by mean free
path (λ). It is calculated by finding statistical average distance the molecules travel between
two successive collisions. If the mean free path is very small as compared with some
characteristic length in the flow domain (i.e., the molecular density is very high) then the gas
can be treated as a continuous medium. If the mean free path is large in comparison to some
characteristic length, the gas cannot be considered continuous and it should be analyzed by the
molecular theory.
A dimensionless parameter known as Knudsen number, Kn = λ / L, where λ is the mean free
path and L is the characteristic length. It describes the degree of departure from continuum.
Usually when Kn> 0.01, the concept of continuum does not hold good.
In this, Kn is always less than 0.01 and it is usual to say that the fluid is a continuum.
Other factor which checks the validity of continuum is the elapsed time between collisions. The
time should be small enough so that the random statistical description of molecular activity
holds good.
In continuum approach, fluid properties such as density, viscosity, thermal conductivity,
temperature, etc. can be expressed as continuous functions of space and time.
The Scale of Pressure
Gauge Pressure
Absolute
Pressure
Vacuum Pressure
Local
atmospheric
Pressure
Absolute Pressure
Absolute Zero
(complete vacuum)
At sea-level, the international standard atmosphere has been chosen as Patm = 101.325 kN/m2
Page 4 of 265
Introduction
Chapter 1
Some special units for Thermodynamics
kPa m 3 /kg
Note: Physicists use below units
Universal gas constant, Ru= 8.314 kJ/kmole − K
Characteristic gas constant, Rc =
For Air R =
Ru
M
8.314 kJ/kmole- K
=
29
kg/kmole
= 0.287 kJ/kg- K
For water R =
8.314 kJ/kmole-K
18
kg/kmole
= 0.461 kJ/kg -K
Units of heat and work is kJ
Units of pressure is kPa
1 atm = 101.325 kPa
1 bar = 100 kPa
1 MPa =1000 kPa.
Page 5 of 265
Introduction
Chapter 1
Questions with Solution P. K. Nag
Q1.1
Solution:
A pump discharges a liquid into a drum at the rate of 0.032 m3/s. The
drum, 1.50 m in diameter and 4.20 m in length, can hold 3000 kg of the
liquid. Find the density of the liquid and the mass flow rate of the liquid
handled by the pump.
(Ans. 12.934 kg/s)
Volume of drum =
=
πd 2
×h
4
π ×1.502
× 4.2 m3
4
= 7.422 m3
mass
3000 kg
=
= 404.203 kg 3
m3
m
Volume 7.422
mass flow rate = Vloume flow rate × density
density =
= 0.032 × 404.203 kg
= 12.9345 kg
Q1.2
s
s
The acceleration of gravity is given as a function of elevation above sea
level by
−6
g = 980.6 – 3.086 × 10 H
Where g is in cm/s2 and H is in cm. If an aeroplane weighs 90,000 N at
sea level, what is the gravity force upon it at 10,000 m elevation? What is
the percentage difference from the sea-level weight?
(Ans. 89,716.4 N, 0.315%)
Solution:
g´ = 980.6 − 3.086 × 10−6 × 10,000 × 100
= 977.514 cm
= 9.77514 m 2
s2
s
90,000
Wsea = 90,000 N =
kgf
9.806
= 9178.054 kgf
Wete = 9178.054 × 9.77514 N = 89716.765 N
90,000 − 89716.765
% less =
× 100%
90,000
= 0.3147% ( less )
Q1.3
Solution:
Prove that the weight of a body at an elevation H above sea-level is given
by
2
mg ⎛ d ⎞
W =
g0 ⎜⎝ d + 2H ⎟⎠
Where d is the diameter of the earth.
According to Newton’s law of gravity it we place a man of m at an height of H
then
Page 6 of 265
Introduction
Chapter 1
Force of attraction =
(i)
GMm
(d 2 + H)
go =
or
Weight ( W ) =
(d 2 + H)
mg ( d )
2
=
(d 2 + H)
GMm
( 2)
d
2
d
= mg o
GM
(d 2 )
2
1.3
2
2
from equation... ( i )
2
⎛ d ⎞
= mg o ⎜
⎟
⎝ d + 2H ⎠
Solution:
H
GMm
o
Q1.4
m
If we place it in a surface of earth
then
Force of attraction =
∴
…
2
2
Pr oved.
The first artificial earth satellite is reported to have encircled the earth
at a speed of 28,840 km/h and its maximum height above the earth’s
surface was stated to be 916 km. Taking the mean diameter of the earth
to be 12,680 km, and assuming the orbit to be circular, evaluate the value
of the gravitational acceleration at this height.
The mass of the satellite is reported to have been 86 kg at sea-level.
Estimate the gravitational force acting on the satellite at the operational
altitude.
(Ans. 8.9 m/s2; 765 N)
Their force of attraction = centrifugal force
Centirfugal force =
mv 2
r
2
⎛ 28840 × 1000 ⎞
86 × ⎜
⎟
60 × 60
⎝
⎠
=
N
⎛ 12680 × 103
3⎞
+ 916 × 10 ⎟
⎜
2
⎝
⎠
= 760.65 N (Weight)
Q1.5
Solution:
Convert the following readings of pressure to kPa, assuming that the
barometer reads 760 mmHg:
(a) 90 cmHg gauge
(b) 40 cmHg vacuum
(c) 1.2 m H2O gauge
(d) 3.1 bar
760 mm Hg = 0.760 × 13600 × 9.81 Pa
= 10139.16 Pa
101.4 kPa
Page 7 of 265
Introduction
Chapter 1
Q1.6
Solution:
Q1.7
(a)
90 cm Hg gauge
= 0.90 × 13600 × 9.81 × 10-3 + 101.4 kPa
= 221.4744 kPa
(b)
40 cm Hg vacuum
= (76 – 40) cm (absolute)
= 0.36 × 43.600 × 9.81 kPa
= 48.03 kPa
(c)
1.2 m H2O gauge
= 1.2 × 1000 × 9.81 × 10-3 + 101.4 kPa
= 113.172 kPa
(d)
3.1 bar = 3.1 × 100 kPa = 310 kPa
A 30 m high vertical column of a fluid of density 1878 kg/m3 exists in a
place where g = 9.65 m/s2. What is the pressure at the base of the column.
(Ans. 544 kPa)
p = z ρg
= 30 × 1878 × 9.65 Pa
= 543.681 kPa
Assume that the pressure p and the specific volume v of the atmosphere
are related according to the equation pv1.4 = 2.3 × 105 , where p is in N/m2
abs and v is in m3/kg. The acceleration due to gravity is constant at 9.81
m/s2. What is the depth of atmosphere necessary to produce a pressure of
l.0132 bar at the earth’s surface? Consider the atmosphere as a fluid
column.
(Ans. 64.8 km)
Page 8 of 265
Introduction
Chapter 1
Solution:
dp = dh ρg
or
dp = dh ×
or
v=
1.4
pv
Zero line
1
×g
v
h
g dh
dp
p = hρg
3
= 2.3 ×10 = 2300
1
or
⎛ 2300 ⎞1.4 ⎛ 2300 ⎞
v=⎜
⎟ =⎜ p ⎟
⎝ p ⎠
⎝
⎠
g dh ⎛ 2300 ⎞
=⎜
⎟
dp
⎝ p ⎠
or
⎛ 2300 ⎞
g dh = ⎜
⎟ dp
⎝ p ⎠
p
n
1
where n =
1.4
dh
p + dp
H
∫ dh =
or
0
h =
or
101320
2300n
g
∫
0
dp
pn
n
2300 ⎡
1−n
(101320 )( ) − 0 ⎤⎦ = 2420 m = 2.42 km
g (1 − n ) ⎣
Q1.8
The pressure of steam flowing in a pipeline is
measured with a mercury manometer, shown in
Figure. Some steam condenses into water. Estimate
the steam pressure in kPa. Take the density of
mercury as 13.6 × 103 kg/m 3 , density of water as 103
kg/m3, the barometer reading as 76.1 cmHg, and g as
9.806 m/s2.
Solution:
po + 0.50 × ρ
Solution:
h
p = hρg
n
Q1.9
dh
n
or
or
HO-h
Hg
× g = 0.03 × ρH2 O × g + p
p = 0.761 × 13.6 × 103 × 9.806 + 0.5 × 13.6 × 103 × 9.806 − 0.03 × 1000 × 9.806 Pa.
= 167.875 kPa
A vacuum gauge mounted on a condenser reads 0.66 mHg. What is the
absolute pressure in the condenser in kPa when the atmospheric
pressure is 101.3 kPa?
(Ans. 13.3 kPa)
Absolute
= atm. – vacuum
= 101.3 – 0.66 × 13.6 × 103 × 9.81 × 10−3 kPa
= 13.24 kPa
Page 9 of 265
Page 10 of 265
Temperature
Chapter 2
2. Temperature
Some Important Notes
Comparison of Temperature scale
100o C
Boiling Point
Test
C
Temperature
0o C
Q2.1
373K
F
Freezing Point
Relation:
212oF
32o F
80 o
x
K
2 73 K
30 cm
0o
10 cm
C−0
F − 32
K − 273
ρ −0
x − 10
=
=
=
=
100 − 0 212 − 32 373 − 273 80 − 0 30 − 10
Questions with Solution P. K. Nag
The limiting value of the ratio of the pressure of gas at the steam point and at
the triple point of water when the gas is kept at constant volume is found to be
1.36605. What is the ideal gas temperature of the steam point?
(Ans. 100°C)
p
= 1.36605
pt
Solution:
∴
θ( v ) = 273.16 ×
p
pt
= 273.16 × 1.36605
= 373.15°C
Q2.2
In a constant volume gas thermometer the following pairs of pressure
readings were taken at the boiling point of water and the boiling point
of sulphur, respectively:
Water b.p.
Sulphur b.p.
50.0
96.4
100
193
200
387
300
582
The numbers are the gas pressures, mm Hg, each pair being taken with
the same amount of gas in the thermometer, but the successive pairs
being taken with different amounts of gas in the thermometer. Plot the
ratio of Sb.p.:H2Ob.p. against the reading at the water boiling point, and
extrapolate the plot to zero pressure at the water boiling point. This
Page 11 of 265
Temperature
Chapter 2
Sb.p
Ratio
= 1.928 1.93
Wb.p
∴
1.926
Extrapolating
Solution :
gives the ratio of Sb.p. : H2Ob.p. On a gas thermometer operating at zero
gas pressure, i.e., an ideal gas thermometer. What is the boiling point of
sulphur on the gas scale, from your plot?
(Ans. 445°C)
Water b.p.
50.0 100
200
300
Sulphur b.p. 96.4 193
387
582
1.935 1.940
T1 = 100°C = 373K
0
T2 = ?
50
100
200
300
p1
= 1.926
p2
∴
Q2.3
T2 = 373 × 1.926 = 718K = 445°C
The resistance of a platinum wire is found to be 11,000 ohms at the ice point,
15.247 ohms at the steam point, and 28.887 ohms at the sulphur point. Find the
constants A and B in the equation
R = R0 (1 + At + Bt2 )
And plot R against t in the range 0 to 660°C.
Solution:
(3271, 1668628)
R
36.595
11
y
0
x
R 0 = 11.000 Ω
{
R100 = R 0 1 + A × 100 + B × 1002
t
660°C
}
4
15.247 = 11.000 + 1100A + 11 × 10 B
or
... ( i )
or 3.861 × 10−3 = A + 100B
28.887 = 11.00 + 445 × 11A + 4452 × 11B
... ( ii )
3.6541×10-3 = A + 445B
equation ( ii ) −
( i ) gives.
B = − 6 × 10 −7
A = 3.921 × 10 −3
{
}
)
R = 11 1 + 3.921 × 10 −3 t − 6 × 10 −7 t 2
or
(
Y = 11 1 + 3.921 × 10 −3 t − 6 × 10 −7 t 2
or ( t − 3271) = − 4 × 37922 × ( Y − 1668628 )
2
R 660 = 36.595
Page 12 of 265
Temperature
Chapter 2
Q2.4
when the reference junction of a thermocouple is kept at the ice point
and the test junction is at the Celsius temperature t, and e.m.f. e of the
thermocouple is given by the equation
ε = at + bt2
Where a = 0.20 mV/deg, and b = - 5.0 × 10-4 mV/deg2
(a)
(b)
Compute the e.m.f. when t = - l00°C, 200°C, 400°C, and 500°C, and
draw graph of ε against t in this range.
Suppose the e.m.f. ε is taken as a thermometric property and that a
temperature scale t* is defined by the linear equation.
t* = a' ε + b'
Solution:
Q2.5
And that t* = 0 at the ice point and t* = 100 at the steam point. Find
the numerical values of a' and b' and draw a graph of ε against t*.
(c) Find the values of t* when t = -100°C, 200°C, 400°C, and 500°C, and
draw a graph of t* against t.
(d) Compare the Celsius scale with the t* scale.
Try please
The temperature t on a thermometric scale is defined in terms of a
property K by the relation
t = a ln K + b
Solution:
Where a and b are constants.
The values of K are found to be 1.83 and 6.78 at the ice point and the
steam point, the temperatures of which are assigned the numbers 0 and
100 respectively. Determine the temperature corresponding to a
reading of K equal to 2.42 on the thermometer.
(Ans. 21.346°C)
t = a ln x + b
0 = a x ln 1.83 + b
… (i)
100 = a x ln 6.78 + b
… (ii)
Equation {(ii) – (i)} gives
or
∴
∴
∴
Q2.6
⎛ 6.78 ⎞
a ⋅ ln⋅ ⎜
⎟ = 100
⎝ 1.83 ⎠
a = 76.35
b = − a × ln 1.83
= − 46.143
t = 76.35 ln k − 46.143
t* = 76.35 × ln 2.42 − 46.143
= 21.33°C
The resistance of the windings in a certain motor is found to be 80 ohms
at room temperature (25°C). When operating at full load under steady
state conditions, the motor is switched off and the resistance of the
windings, immediately measured again, is found to be 93 ohms. The
windings are made of copper whose resistance at temperature t°C is
given by
Page 13 of 265
Temperature
Chapter 2
Rt = R0 [1 + 0.00393 t]
Solution:
Where R0 is the resistance at 0°C. Find the temperature attained by the
coil during full load.
(Ans. 70.41°C)
R25 = R0 [1 + 0.00393 × 25]
∴
Q2.7
R0 =
80
= 72.84 Ω
1
+
0.00393
× 25]
[
∴
93 = 72.84 {1 + 0.00393 × t}
or
t = 70.425°C
A new scale N of temperature is divided in such a way that the freezing
point of ice is 100°N and the boiling point is 400°N. What is the
temperature reading on this new scale when the temperature is 150°C?
At what temperature both the Celsius and the new temperature scale
reading would be the same?
(Ans. 550°N, – 50°C.)
Solution:
150 − 0
N − 100
=
100 − 0
400 − 100
or N = 550o N
let N= C for x o
C −0
N − 100
then
=
100 − 0
400 − 100
x
x − 100
=
or
300
100
or
or
or
or
Q2.8
x
x − 100
3
3 x = x -100
2 x = -100
x = - 50o C
=
A platinum wire is used as a resistance thermometer. The wire resistance was
found to be 10 ohm and 16 ohm at ice point and steam point respectively, and
30 ohm at sulphur boiling point of 444.6°C. Find the resistance of the wire at
500°C, if the resistance varies with temperature by the relation.
R = R0 (1 + α t + β t2 )
(Ans. 31.3 ohm)
Solution:
10 = R0 (1 + 0 × α + β × 02 )
16 = R0 (1 + 100 × α + β × 1002 )
30 = R0 (1 + α × 444.6 + β × 444.62 )
Solve R0 ,α & β then
R = R0 (1 + 500 × α + β × 5002 )
Page 14 of 265
Work and Heat Transfer
Chapter 3
3.
Work and Heat Transfer
Some Important Notes
-ive
W
+ive
W
+ive
Q
-ive
Q
Our aim is to give heat to the system and gain work output from it.
So heat input → +ive (positive)
Work output → +ive (positive)
f
vf
i
vi
Wi− f = ∫ pdV = ∫ pdv
d Q = du + dW
f
f
∫ dQ = uf − ui + ∫ dW
i
i
vf
Qi− f = uf − ui + ∫ pdV
vi
Questions with Solution P. K. Nag
Q3.1
(a)A pump forces 1 m3/min of water horizontally from an open well to a closed
tank where the pressure is 0.9 MPa. Compute the work the pump must do
upon the water in an hour just to force the water into the tank against the
pressure. Sketch the system upon which the work is done before and after
the process.
(Ans. 5400 kJ/h)
(b)If the work done as above upon the water had been used solely to raise the
same amount of water vertically against gravity without change of
pressure, how many meters would the water have been elevated?
(Ans. 91.74 m)
(c)If the work done in (a) upon the water had been used solely to accelerate
the water from zero velocity without change of pressure or elevation, what
velocity would the water have reached? If the work had been used to
accelerate the water from an initial velocity of 10 m/s, what would the final
velocity have been?
(Ans. 42.4 m/s; 43.6 m/s)
Solution:
(a)
Flow rate 1m3/hr.
Pressure of inlet water = 1 atm = 0.101325 MPa
Pressure of outlet water = 0.9 MPa
Page 15 of 265
Work and Heat Transfer
Chapter 3
∴
Power = Δpv
= ( 0.9 − 0.101325 ) × 103 kPa ×
= 13.31 kJ
(b)
1 m3
s
60
s
So that pressure will be 0.9 MPa
∴
hρg = 0.9 MPa
or
h=
0.9 × 106
m = 91.743 m
1000 × 9.81
1
V22 − V12 = Δpv
m
2
(
(c)
or
or
or
)
= v ρ
where m
1
ρ V22 − V12 = Δp
2
(
)
Δp
ρ
Δp
V22 = V12 + 2
ρ
V22 − V12 = 2
= 102 +
2 × ( 0.9 − 0.101325 ) × 106
1000
V2 = 41.2 m / s.
Q3.2
The piston of an oil engine, of area 0.0045 m2, moves downwards 75 mm,
drawing in 0.00028 m3 of fresh air from the atmosphere. The pressure in the
cylinder is uniform during the process at 80 kPa, while the atmospheric
pressure is 101.325 kPa, the difference being due to the flow resistance in the
induction pipe and the inlet valve. Estimate the displacement work done by
the air finally in the cylinder.
(Ans. 27 J)
Solution : Volume of piston stroke
-4
Final volume = 3.375×10 m3
= 0.0045 × 0.075m3
= 0.0003375m3
∴ ΔV = 0.0003375 m3
as pressure is constant
= 80 kPa
So work done = pΔV
= 80 × 0.0003375 kJ
Initial volume = 0
= 0.027 kJ = 27 J
Q3.3
Solution:
An engine cylinder has a piston of area 0.12 m3 and contains gas at a
pressure of 1.5 MPa. The gas expands according to a process which is
represented by a straight line on a pressure-volume diagram. The final
pressure is 0.15 MPa. Calculate the work done by the gas on the piston if
the stroke is 0.30 m.
(Ans. 29.7 kJ)
Initial pressure ( p1 ) = 1.5 MPa
Final volume (V1) = 0.12m2 × 0.3m
Page 16 of 265
Work and Heat Transfer
Chapter 3
= 0.036 m3
Final pressure ( p2 ) = 0.15 MPa
As initial pressure too high so the volume is neglected.
Work done = Area of pV diagram
1
( p1 + p2 ) × V
2
1
= (1.5 + 0.15 ) × 0.036 × 103 kJ
2
= 29.7 kJ
=
p
1.5 MPa
0.15 MPa
neg.
V
0.36 m3
Q3.4
Solution:
A mass of 1.5 kg of air is compressed in a quasi-static process from 0.1
MPa to 0.7 MPa for which pv = constant. The initial density of air is 1.16
kg/m3. Find the work done by the piston to compress the air.
(Ans. 251.62 kJ)
For quasi-static process
Work done =
∫ pdV
[ given pV = C
v2
dV
V
v1
= p1 V1 ∫
∴ p1 V1 = pV = p 2 V2 = C
⎛V ⎞
= p1 V1 l n ⎜ 2 ⎟
⎝ V1 ⎠
∴
p=
⎛p ⎞
= p1 V1 ln ⎜ 1 ⎟
⎝ p2 ⎠
∴
p1 V2
=
p2 V1
= 0.1 × 1.2931 × ln
0.1
MJ
0.7
= 251.63 kJ
p1 V1
V
given p1 = 0.1 MPa
V1 =
m1
1.5
=
m3
ρ1
1.16
p2 = 0.7 MPa
Q3.5
Solution:
A mass of gas is compressed in a quasi-static process from 80 kPa, 0.1 m3
to 0.4 MPa, 0.03 m3. Assuming that the pressure and volume are related
by pvn = constant, find the work done by the gas system.
(Ans. –11.83 kJ)
Given initial pressure ( p1 ) = 80kPa
Initial volume ( V1 ) = 0.1 m3
Page 17 of 265
Work and Heat Transfer
Chapter 3
Final pressure ( p2 ) = 0.4 MPa = 400 kPa
Final volume ( V2 ) = 0.03 m3
As p-V relation pV n = C
∴
p1 V1n = p2 V2n
taking log e both side
ln p1 + n ln V1 = ln p2 + n ln V2
or
n [ ln V1 − ln V2 ] = ln p2 − ln p1
or
⎛V ⎞
⎛p ⎞
n ln ⎜ 1 ⎟ = ln ⎜ 2 ⎟
⎝ V2 ⎠
⎝ p1 ⎠
or
p
ln ⎛⎜ 2 ⎞⎟
⎝ p1 ⎠
n=
V
ln ⎛⎜ 1 ⎞⎟
⎝ V2 ⎠
∴
Q3.6
Solution:
⎛ 400 ⎞
ln ⎜
⎟
⎝ 80 ⎠ = 1.60944 ≈ 1.3367 ≈ 1.34
=
1.20397
⎛ 0.1 ⎞
ln ⎜
⎟
⎝ 0.03 ⎠
p V − p2 V2
Work done ( W ) = 1 1
n −1
80 × 0.1 − 400 × 0.03
=
= − 11.764 kJ
1.34 − 1
A single-cylinder, double-acting, reciprocating water pump has an
indicator diagram which is a rectangle 0.075 m long and 0.05 m high. The
indicator spring constant is 147 MPa per m. The pump runs at 50 rpm.
The pump cylinder diameter is 0.15 m and the piston stroke is 0.20 m.
Find the rate in kW at which the piston does work on the water.
(Ans. 43.3 kW)
−3
2
2
Area of indicated diagram ( ad ) = 0.075 × 0.05 m = 3.75 × 10 m
Spring constant (k) = 147 MPa/m
Page 18 of 265
Work and Heat Transfer
Chapter 3
Q3.7
Solution:
A single-cylinder, single-acting, 4 stroke engine of 0.15 m bore develops
an indicated power of 4 kW when running at 216 rpm. Calculate the area
of the indicator diagram that would be obtained with an indicator
having a spring constant of 25 × 106 N/m3. The length of the indicator
diagram is 0.1 times the length of the stroke of the engine.
(Ans. 505 mm2)
Given Diameter of piston (D) = 0.15 m
I.P = 4 kW = 4 × 1000 W
Speed (N) = 216 rpm
Spring constant (k) = 25 × 106 N/m
Length of indicator diagram ( l d ) = 0.1 × Stoke (L)
Let Area of indicator diagram = ( ad )
∴
Mean effective pressure ( pm ) =
and
∴
or
pm LAN
[as 4 stroke engine]
120
a ×k L×A×N
I.P. = d
×
ld
120
I.P. =
ad =
or
ad
×k
ld
I.P × l d × 120
k×L×A×N
⎡
πD2 ⎤
=
area
A
⎢
⎥
4 ⎥
⎢
⎣⎢and l d = 0.1L ⎦⎥
I.P × 0.1 L × 120 × 4
=
k × L × π × D2 × N
4 × 0.1 × 120 × 4 × 1000
m2
25 × 106 × π × 0.152 × 216
= 5.03 × 10−4 m2
=
= 503 mm2
Q3.8
Solution:
A six-cylinder, 4-stroke gasoline engine is run at a speed of 2520 RPM.
The area of the indicator card of one cylinder is 2.45 × 103 mm2 and its
length is 58.5 mm. The spring constant is 20 × 106 N/m3. The bore of the
cylinders is 140 mm and the piston stroke is 150 mm. Determine the
indicated power, assuming that each cylinder contributes an equal
power.
(Ans. 243.57 kW)
a
pm = d × k
ld
2.45 × 103
× 20 × 103 Pa
58.5
= 837.607 kPa
=
∴
mm2 N
mm × N ⎛ 1 ⎞
2
× 3 ⇒
=⎜
⎟N / m
mm m
1000
m × m2
⎝
⎠
L = 0.150 m
Page 19 of 265
Work and Heat Transfer
Chapter 3
πD2
π × 0.142
=
4
4
N = 2520
n=6
A =
∴
I.P. =
pm LAN
×n
120
= 837.607 × 0.15 ×
[as four stroke]
π × 0.142 2520 × 6
×
kW
4
120
= 243.696 kW
Q3.9
Solution:
A closed cylinder of 0.25 m diameter is fitted with a light frictionless
piston. The piston is retained in position by a catch in the cylinder wall
and the volume on one side of the piston contains air at a pressure of 750
kN/m2. The volume on the other side of the piston is evacuated. A helical
spring is mounted coaxially with the cylinder in this evacuated space to
give a force of 120 N on the piston in this position. The catch is released
and the piston travels along the cylinder until it comes to rest after a
stroke of 1.2 m. The piston is then held in its position of maximum travel
by a ratchet mechanism. The spring force increases linearly with the
piston displacement to a final value of 5 kN. Calculate the work done by
the compressed air on the piston.
(Ans. 3.07 kJ)
Work done against spring is work done by the compressed gas
φ 0.25m
1.2m
120 + 5000
2
= 2560 N
Travel = 1.2 m
∴ Work Done = 2560 × 1.2 N.m
= 3.072 kJ
By Integration
At a travel (x) force (Fx) = 120 + kx
At 1.2 m then 5000 = 120 + k × 1.2
∴
Fx = 120 + 4067 x
Mean force =
Page 20 of 265
Work and Heat Transfer
Chapter 3
1.2
∴
W=
∫ F dx
x
0
1.2
=
∫ [120 + 4067x ] dx
0
1.2
⎡
x2 ⎤
= ⎢120x + 4067 × ⎥
2 ⎦0
⎣
= 120 × 1.2 + 4067 ×
1.22
J
2
= 144 + 2928.24 J
= 3072.24J = 3.072 kJ
Q 3.l0
A steam turbine drives a ship’s propeller through an 8: 1 reduction gear.
The average resisting torque imposed by the water on the propeller is
750 × 103 mN and the shaft power delivered by the turbine to the
reduction gear is 15 MW. The turbine speed is 1450 rpm. Determine (a)
the torque developed by the turbine, (b) the power delivered to the
propeller shaft, and (c) the net rate of working of the reduction gear.
(Ans. (a) T = 98.84 km N, (b) 14.235 MW, (c) 0.765 MW)
Solution:
Power of the propeller = Power on turbine shaft
The net rate of working of the reduction gear
= (15 – 14.235) MW
= 0.7647 MW
Q 3.11
A fluid, contained in a horizontal cylinder fitted with a frictionless leak
proof piston, is continuously agitated by means of a stirrer passing
through the cylinder cover. The cylinder diameter is 0.40 m. During the
stirring process lasting 10 minutes, the piston slowly moves out a
distance of 0.485 m against the atmosphere. The net work done by the
fluid during the process is 2 kJ. The speed of the electric motor driving
the stirrer is 840 rpm. Determine the torque in the shaft and the power
output of the motor.
(Ans. 0.08 mN, 6.92 W)
Page 21 of 265
Work and Heat Transfer
Chapter 3
Solution:
Change of volume = A L
πd 2
×L
4
π × 0.4 2
=
× 0.485 m3
4
= 0.061 m3
=
As piston moves against constant atmospheric pressure then work done = pΔV
φ = 0.4m
M
0.485m
= 101.325 × 0.061 kJ
= 6.1754 kJ
Net work done by the fluid = 2 kJ
∴ Net work done by the Motor = 4.1754 kJ
There for power of the motor
4.1754 × 103
W
10 × 60
= 6.96 W
P
Torque on the shaft =
W
6.96 × 60
=
2π × 840
=
= 0.0791mN
Q3.12
At the beginning of the compression stroke of a two-cylinder internal
combustion engine the air is at a pressure of 101.325 kPa. Compression
reduces the volume to 1/5 of its original volume, and the law of
compression is given by pv1.2 = constant. If the bore and stroke of each
cylinder is 0.15 m and 0.25 m, respectively, determine the power
absorbed in kW by compression strokes when the engine speed is such
that each cylinder undergoes 500 compression strokes per minute.
(Ans. 17.95 kW)
Page 22 of 265
Work and Heat Transfer
Chapter 3
Solution:
πd 2
×L
4
2
π × ( 0.15 )
=
× 0.25 m3
4
= 0.00442 m3
Initial volume ( V1 ) =
Initial p r essure ( p1 ) = 101.325 kPa.
V1
= 0.000884 m3
5
= p2 V21.2
Final volume ( V2 ) =
p1 V11.2
p2 =
Or
p1 V11.2
= 699.41 ≈ 700 kPa
V21.2
Work done / unit stroke − unit cylinder ( W )
⎛ 1.2 ⎞
=⎜
⎟ × [ p1 V1 − p2 V2 ]
⎝ 1.2 − 1 ⎠
⎛ 101.325 × 0.00442 − 700 × 0.000884 ⎞
=⎜
⎟ × 1.2
1.2 − 1
⎝
⎠
-ive
work,
as
work
done
on the system )
(
W × 500 × 2 × 1.2
kW
60
= 17.95 kW
Power =
Q3.13
Determine the total work done by a gas system following an expansion
process as shown in Figure.
(Ans. 2.253 MJ)
Solution:
Area under AB
= (0.4 – 0.2) × 50 × 105 J
= 10
6
W = 1 MJ
Page 23 of 265
Work and Heat Transfer
Chapter 3
A
bar
p
50
B
pV1.3 = c
C
0.2
0.4
0.8
V1 m3
Area under BC
p V − p2 V2
= 1 1
n −1
50 × 105 × 0.4 − 20.31 × 105 × 0.8
W
=
1.3 − 1
= 1.251MJ
⎡
⎢ Here
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
pB = pB = 50 bar = 50 × 105 Pa
VB = 0.4m3
VC = 0.8m3
pC =
pB VB1.3
VC1.3
=
50 × 105 × 0.41.3
0.81.3
= 20.31 × 105 Pa
Total work = 2.251MJ
Q3.14
A system of volume V contains a mass m of gas at pressure p and
temperature T. The macroscopic properties of the system obey the
following relationship:
a ⎞
⎛
⎜ p + 2 ⎟ (V − b) = mRT
V ⎠
⎝
Solution:
Where a, b, and R are constants.
Obtain an expression for the displacement work done by the system
during a constant-temperature expansion from volume V1 to volume V2.
Calculate the work done by a system which contains 10 kg of this gas
expanding from 1 m3 to 10 m3 at a temperature of 293 K. Use the values
a = 15.7 × 10 Nm 4 , b = 1.07 × 10−2 m 3 , and R = 0.278 kJ/kg-K.
(Ans. 1742 kJ)
As it is constant temp-expansion then
a ⎞
⎛
⎜ p + 2 ⎟ ( V − b ) = constant ( mRT ) ( k ) as T = constant
V ⎠
⎝
Page 24 of 265
Work and Heat Transfer
Chapter 3
⎛
⎛
a ⎞
a ⎞
⎜ p1 + 2 ⎟ ( V1 − b ) = ⎜ p2 + 2 ⎟ ( V2 − b ) = ( k )
V1 ⎠
V2 ⎠
⎝
⎝
∴
2
W = ∫ p dV
∴
1
a ⎞ constant ( k )
⎛
⎜p+ V ⎟ =
V−b
⎝
⎠
2
a ⎞
⎛ k
= ∫⎜
− 2 ⎟ dV
V−b V ⎠
1⎝
or
2
a⎤
⎡
= ⎢ k ln ( V − b ) + ⎥
V
⎣
⎦1
p=
−∫
k
a
− 2
V−b V
1
1
dv = + c
V2
V
⎛ V − b⎞
⎛ 1
1 ⎞
= k ln ⎜ 2
−
⎟ + a⎜
⎟
⎝ V1 − b ⎠
⎝ V2 V1 ⎠
⎡⎛
⎛ 1
V −b
a ⎞
1 ⎞⎤
= ⎢⎜ p1 + 2 ⎟ ( V1 − b ) ln 2
+ a⎜
−
⎟⎥
−
V
V
b
V
1 ⎠
1
⎝ 2 V1 ⎠ ⎦
⎣⎝
a ⎞
⎛
⎜ p + 2 ⎟ ( V − b ) = constant ( mRT ) ( k ) as T = constant
V
⎝
⎠
Given m = 10 kg; T = 293 K; R = 0.278 kJ/kg. K
∴
Constant k = 10 × 293 × 0.278 kJ = 814.54 kJ
a = 15.7 × 10 Nm4; b = 1.07 × 10-2m3
⇒ V2 = 10m3, V1 = 1m3
∴
⎛ 10 − 1.07 × 10−2 ⎞
⎛ 1 1⎞
+ a⎜ − ⎟
W = 814.54 ln ⎜
−2 ⎟
⎝ 10 1 ⎠
⎝ 1 − 1.07 × 10 ⎠
= (1883.44 − a × 0.9 ) kJ
= (1883.44 − 157 × 0.9 ) kJ
= 1742.14 kJ
Q3.15
Solution:
If a gas of volume 6000 cm3 and at pressure of 100 kPa is compressed
quasistatically according to pV2 = constant until the volume becomes
2000 cm3, determine the final pressure and the work transfer.
(Ans. 900 kPa, – 1.2 kJ)
Initial volume ( v1 ) = 6000 cm3
= 0.006 m3
Initial pressure ( p1 ) = 100 kPa
Final volume ( v 2 ) = 2000 cm3
= 0.002 m3
If final pressure ( p2 )
p V 2 100 × ( 0.006 )
p2 = 1 21 =
= 900 kPa
2
V2
( 0.002 )
2
∴
Page 25 of 265
Work and Heat Transfer
Chapter 3
work done on the system =
1
⎡p2 V2 − p1 V1 ⎤⎦
n −1 ⎣
1
⎡900 × 0.002 − 100 × 0.006⎤⎦ kJ
2 −1⎣
= 1.2 kJ
=
Q3.16
Solution:
The flow energy of 0.124 m3/min of a fluid crossing a boundary to a
system is 18 kW. Find the pressure at this point.
(Ans. 8709 kPa)
If pressure is p1
Area is A1
Velocity is V1
Volume flow rate (Q) = A1V1
∴
Power = force × velocity
= p1A1 × V1
p1
= p × (Q)
1
∴
or
Q3.17
Solution:
0.124
60
18 × 60
p1 =
kPa
0.124
= 8.71 MPa
V1
18 = p1 ×
A1
A milk chilling unit can remove heat from the milk at the rate of 41.87
MJ/h. Heat leaks into the milk from the surroundings at an average rate
of 4.187 MJ/h. Find the time required for cooling a batch of 500 kg of
milk from 45°C to 5°C. Take the cp of milk to be 4.187 kJ/kg K.
(Ans. 2h 13 min)
Heat to be removed (H) = mst
= 500 × 4.187 × (45-5) kJ
= 83.740 MJ
Net rate of heat removal
−H
=H
rej
leak
= ( 41.87 − 4.187 ) MJ / h
Q3.18
Solution:
= 37.683 MJ / h
83.740
∴
Time required =
hr
37.683
= 2 hr. 13 min . 20 sec .
680 kg of fish at 5°C are to be frozen and stored at – 12°C. The specific
heat of fish above freezing point is 3.182, and below freezing point is
1.717 kJ/kg K. The freezing point is – 2°C, and the latent heat of fusion is
234.5 kJ/kg. How much heat must be removed to cool the fish, and what
per cent of this is latent heat?
(Ans. 186.28 MJ, 85.6%)
Heat to be removed above freezing point
= 680 × 3.182 × {5 – (-2)} kJ
= 15.146 MJ
Page 26 of 265
Work and Heat Transfer
Chapter 3
Heat to be removed latent heat
= 680 × 234.5 kJ
= 159.460 MJ
Heat to be removed below freezing point
= 680 × 1.717 × {– 2 – (– 12)} kJ
= 11.676 MJ
∴
Total Heat = 186.2816 MJ
% of Latent heat =
159.460
× 100 = 85.6 %
186.2816
Page 27 of 265
Page 28 of 265
First Law of Thermodynamics
Chapter 4
4.
First Law of Thermodynamics
Some Important Notes
•
dQ is an inexact differential, and we write
∫
2
1
•
dQ = Q1−2 or
Q2 ≠ Q2 − Q1
dW is an inexact differential, and we write
W1−2 =
•
1
(ΣQ)cycle = (ΣW)cycle
or
v∫ δ Q
=
∫
2
1
dW =
∫
2
1
pdV ≠ W2 − W1
v∫ δ W
The summations being over the entire cycle.
•
•
•
•
•
•
δQ – δW = dE
An isolated system which does not interact with the surroundings Q = 0 and W = 0.
Therefore, E remains constant for such a system.
The Zeroth Law deals with thermal equilibrium and provides a means for measuring
temperatures.
The First Law deals with the conservation of energy and introduces the concept of
internal energy.
The Second Law of thermodynamics provides with the guidelines on the conversion heat
energy of matter into work. It also introduces the concept of entropy.
The Third Law of thermodynamics defines the absolute zero of entropy. The entropy of a
pure crystalline substance at absolute zero temperature is zero.
Summation of 3 Laws
• Firstly, there isn’t a meaningful temperature of the source from which we can get the full
conversion of heat to work. Only at infinite temperature one can dream of getting the full 1
kW work output.
• Secondly, more interestingly, there isn’t enough work available to produce 0K. In other
words, 0 K is unattainable. This is precisely the Third law.
Page 29 of 265
First Law of Thermodynamics
Chapter 4
• Because, we don’t know what 0 K looks like, we haven’t got a starting point for the
temperature scale!! That is why all temperature scales are at best empirical.
You can’t get something for nothing:
To get work output you must give some thermal energy.
You can’t get something for very little:
To get some work output there is a minimum amount of thermal energy that
needs to be given.
You can’t get every thing:
However much work you are willing to give 0 K can’t be reached.
Violation of all 3 laws:
Try to get everything for nothing.
Page 30 of 265
First Law of Thermodynamics
Chapter 4
Questions with Solution P. K. Nag
Q4.1
Solution:
An engine is tested by means of a water brake at 1000 rpm. The
measured torque of the engine is 10000 mN and the water consumption
of the brake is 0.5 m3/s, its inlet temperature being 20°C. Calculate the
water temperature at exit, assuming that the whole of the engine power
is ultimately transformed into heat which is absorbed by the cooling
water.
(Ans. 20.5°C)
Power = T.ω
⎛ 2π × 1000 ⎞
= 10000 × ⎜
⎟
60
⎝
⎠
= 1.0472 × 106 W
= 1.0472MW
Let final temperature = t°C
s Δt
∴ Heat absorb by cooling water / unit = m
= v ρs Δt
= 0.5 × 1000 × 4.2 × ( t − 20 )
∴
0.5 × 1000 × 4.2 × ( t − 20 ) = 1.0472 × 10
∴
t − 20 = 0.4986 ≈ 0.5
∴
t = 20.5°C
6
Q4.2
In a cyclic process, heat transfers are + 14.7 kJ, – 25.2 kJ, – 3.56 kJ and +
31.5 kJ. What is the net work for this cyclic process?
Solution :
∑ Q = (14.7 + 31.5 − 25.2 − 3.56 ) kJ
(Ans. 17.34 kJ)
-25.2kJ
= 17.44 kJ
From first law of thermodynamics
(for a cyclic process)
+14.7kJ
∑Q = ∑W
∴ ∑ W = 17.44 kJ
Q4.3
-3.56kJ
31.5kJ
A slow chemical reaction takes place in a fluid at the constant pressure
of 0.1 MPa. The fluid is surrounded by a perfect heat insulator during
the reaction which begins at state 1 and ends at state 2. The insulation is
then removed and 105 kJ of heat flow to the surroundings as the fluid
goes to state 3. The following data are observed for the fluid at states 1, 2
and 3.
State
v (m3)
t (°C)
1
0.003
20
2
0.3
370
3
0.06
20
For the fluid system, calculate E2 and E3, if E1 = 0
(Ans. E2 = – 29.7 kJ, E3 = – 110.7 kJ)
Page 31 of 265
First Law of Thermodynamics
Chapter 4
Solution:
From first law of thermodynamics
dQ = ΔE + pdV
∴
Q = ΔE + ∫ pdV
2
∴
Q1−2 = ( E2 − E1 ) + ∫ pdV
1
or
or
[as insulated Q2−3 = 0]
= ( E2 − E1 ) + 0.1 × 103 (0.3 − 0.003)
E2 = − 29.7 kJ
3
Q2−3 = ( E3 − E2 ) + ∫ pdV
2
or
−105 = ( E3 − E2 ) + 0.1 × 103 ( 0.06 − 0.3 )
or
−105 = E3 + 29.7 + 0.1 × 103 ( 0.06 − 0.3 )
or
−105 = E3 + 29.7 − 24
or
Q4.4
Solution:
E3 = − 105 − 29.7 + 24
= − 110.7 kJ
During one cycle the working fluid in an engine engages in two work
interactions: 15 kJ to the fluid and 44 kJ from the fluid, and three heat
interactions, two of which are known: 75 kJ to the fluid and 40 kJ from
the fluid. Evaluate the magnitude and direction of the third heat
transfer.
(Ans. – 6 kJ)
From first law of thermodynamics
W = -15kJ
∑ dQ = ∑ dW
1
∴
Q1 + Q2 + Q3 = W1 + W2
or
75 − 40 + Q3 = − 15 + 44
Q1 = 75kJ
W2 = 44kJ
Q3 = − 6kJ
i.e. 6kJ from the system
Q = -40kJ
Q4.5
Solution:
Q3
A domestic refrigerator is loaded with food and the door closed. During
a certain period the machine consumes 1 kWh of energy and the internal
energy of the system drops by 5000 kJ. Find the net heat transfer for the
system.
(Ans. – 8.6 MJ)
Q = ΔE + W
Q2 −1 = ( E2 − E1 ) + W2 −1
−1000 × 3600
kJ
1000
= − 8.6MJ
= − 5000kJ +
Page 32 of 265
-W
First Law of Thermodynamics
Chapter 4
Q4.6
Solution:
Q4.7
1.5 kg of liquid having a constant specific heat of 2.5 kJ/kg K is stirred in
a well-insulated chamber causing the temperature to rise by 15°C. Find
Δ E and W for the process.
(Ans. Δ E = 56.25 kJ, W = – 56.25 kJ)
Heat added to the system = 1.5 × 2.5 × 15kJ
= 56.25 kJ
∴
ΔE rise = 56.25kJ
As it is insulated then dQ = 0
∴
ΔQ = ΔE + W
or
0 = 56.25 + W
or
W = – 56.25 kJ
Solution:
The same liquid as in Problem 4.6 is stirred in a conducting chamber.
During the process 1.7 kJ of heat are transferred from the liquid to the
surroundings, while the temperature of the liquid is rising to 15°C. Find
Δ E and W for the process.
(Ans. Δ E = 54.55 kJ, W = 56.25 kJ)
As temperature rise is same so internal energy is same
ΔE = 56.25 kJ
As heat is transferred form the system so we have to give more work = 1.7 kJ to
the system
So
W = – 56.25 – 1.7 kJ
= –57.95 kJ
Q4.8
The properties of a certain fluid are related as follows:
u = 196 + 0.718 t
pv = 0.287 (t + 273)
Solution:
Where u is the specific internal energy (kJ/kg), t is in °C, p is pressure
(kN/m2), and v is specific volume (m3/kg). For this fluid, find cv and cp.
(Ans. 0.718, 1.005 kJ/kg K)
⎛ ∂h ⎞
Cp = ⎜
⎟
⎝ ∂T ⎠ p
⎡ ∂ ( u + pV ) ⎤
=⎢
⎥
∂T
⎣
⎦p
⎡ ∂ {196 + 0.718t + 0.287 ( t + 273 )} ⎤
=⎢
⎥
∂T
⎣⎢
⎦⎥ p
⎡ 0 + 0.718 ∂t + 0.287 ∂t + 0 ⎤
=⎢
⎥
∂T
⎣
⎦p
∂t ⎤
⎡
= ⎢1.005 ⎥
∂T ⎦ p
⎣
= 1.005 kJ / kg − K
Page 33 of 265
⎡ T = t + 273⎤
⎢∴∂T = ∂t ⎥
⎣
⎦
First Law of Thermodynamics
Chapter 4
⎛ ∂u ⎞
cv = ⎜
⎟
⎝ ∂T ⎠ v
⎡ ∂ (196 + 0.718t ) ⎤
=⎢
⎥
∂T
⎣
⎦v
∂t ⎤
⎡
= ⎢0 + 0.718 ⎥
∂T ⎦ v
⎣
= 0.718 kJ / kg − K
Q4.9
Solution:
A system composed of 2 kg of the above fluid expands in a frictionless
piston and cylinder machine from an initial state of 1 MPa, 100°C to a
final temperature of 30°C. If there is no heat transfer, find the net work
for the process.
(Ans. 100.52 kJ)
Heat transfer is not there so
Q = ΔE + W
W = − ΔE
= − ΔU
2
= − ∫ Cv dT
1
= − 0.718 ( T2 − T1 )
= − 0.718 (100 − 30 )
= − 50.26 kJ / kg
∴
Total work (W) = 2 × (-50.26) = -100.52 kJ
Q 4.10
If all the work in the expansion of Problem 4.9 is done on the moving
piston, show that the equation representing the path of the expansion in
the pv-plane is given by pvl.4 = constant.
Solution:
Let the process is pV n = constant.
Then
p V − p2 V2
Work done = 1 1
n −1
mRT 1 − mRT2
=
n −1
=
or
or
or
Q4.11
=
mR
( T1 − T2 )
n −1
2 × 0.287 × (100 − 30 )
n −1
n − 1 = 0.39972
n = 1.39972 ≈ 1.4
[∴
pV = mRT]
⎡R = ( c p − c v )
⎤
⎢
⎥
⎢ = 1.005 − 0.718
⎥
⎢ = 0.287 kJ / kg − K ⎥
⎣⎢
⎦⎥
= 100.52
A stationary system consisting of 2 kg of the fluid of Problem 4.8
expands in an adiabatic process according to pvl.2 = constant. The initial
Page 34 of 265
First Law of Thermodynamics
Chapter 4
conditions are 1 MPa and 200°C, and the final pressure is 0.1 MPa. Find
W and Δ E for the process. Why is the work transfer not equal to ∫ pdV ?
(Ans. W= 217.35, Δ E = – 217.35 kJ,
Solution:
T2 ⎛ p2 ⎞
=⎜ ⎟
T1 ⎝ p1 ⎠
∴
n −1
n
⎛ 0.1 ⎞
=⎜
⎟
⎝ 1 ⎠
∫ pdV
= 434.4 kJ)
1.2 −1
1.2
0.2
T2 = T1 × ( 0.10 )1.2
= 322.251
= 49.25°C
From first law of thermodynamics
dQ = ΔE + dW
∴
∴
0 = ∫ Cv dT + dW
dW = − ∫ Cv dT
2
= − 0.718 × ∫ dT = − 0.718 × ( 200 − 49.25 ) kJ / kg
1
∴
W = − 2× W
= − 2 × 108.2356kJ
= − 216.5kJ
ΔE = 216.5kJ
p V −p V
∫ pdV = 1 n1 − 12 2
mRT1 − mRT2
=
n −1
mR ( T1 − T2 )
=
n −1
2 × 0.287 ( 200 − 49.25 )
=
(1.2 − 1)
= 432.65kJ
As this is not quasi-static process so work is not ∫ pdV .
Q4.12
A mixture of gases expands at constant pressure from 1 MPa, 0.03 m3 to
0.06 m3 with 84 kJ positive heat transfer. There is no work other than
that done on a piston. Find DE for the gaseous mixture.
(Ans. 54 kJ)
The same mixture expands through the same state path while a stirring
device does 21 kJ of work on the system. Find Δ E, W, and Q for the
process.
(Ans. 54 kJ, – 21 kJ, 33 kJ)
Page 35 of 265
First Law of Thermodynamics
Chapter 4
Solution:
Work done by the gas ( W ) = ∫ pdV
= p ( V2 − V1 )
= 1 × 103 ( 0.06 − 0.03 ) kJ
= 30kJ
Heat added = 89kJ
∴
Q = ΔE + W
ΔE = Q − W = 89 − 30 = 54kJ
or
Q4.13
Solution:
A mass of 8 kg gas expands within a flexible container so that the p–v
relationship is of the from pvl.2 = constant. The initial pressure is 1000
kPa and the initial volume is 1 m3. The final pressure is 5 kPa. If specific
internal energy of the gas decreases by 40 kJ/kg, find the heat transfer in
magnitude and direction.
(Ans. + 2615 kJ)
T2 ⎛ p2 ⎞
=⎜ ⎟
T1 ⎝ p1 ⎠
∴
n −1
n
⎛V ⎞
=⎜ 1⎟
⎝ V2 ⎠
p2 ⎛ V1 ⎞
=⎜
⎟
p1 ⎝ V2 ⎠
n −1
n
1
or
V2 ⎛ p1 ⎞ n
=⎜ ⎟
V1 ⎝ p2 ⎠
or
⎛ p ⎞n
V2 = V1 × ⎜ 1 ⎟
⎝ p2 ⎠
1
1
∴
∴
Q4.14
⎛ 1000 ⎞1.2
= 1×⎜
= 82.7 m3
⎟
⎝ 5 ⎠
p V − p2 V2
W= 1 1
n −1
1000 × 1 − 5 × 82.7
=
= 2932.5kJ
1.2 − 1
ΔE = − 8 × 40 = − 320 kJ
Q = ΔE + W = − 320 + 2932.5 = 2612.5kJ
A gas of mass 1.5 kg undergoes a quasi-static expansion which follows a
relationship p = a + bV, where a and b are constants. The initial and final
pressures are 1000 kPa and 200 kPa respectively and the corresponding
volumes are 0.20 m3 and 1.20 m3. The specific internal energy of the gas
is given by the relation
u = l.5 pv – 85 kJ/kg
Where p is the kPa and v is in m3/kg. Calculate the net heat transfer and
the maximum internal energy of the gas attained during expansion.
(Ans. 660 kJ, 503.3 kJ)
Page 36 of 265
First Law of Thermodynamics
Chapter 4
Solution:
1000 = a + b × 0.2
.... ( i )
200 = a + b × 1.2
... ( ii )
( ii ) − ( i ) gives
−800 = b
∴ a = 1000 + 2 × 800 = 1160
∴
p = 1160 − 800V
∴ W=
v2
∫ pdV
v1
1.2
=
∫ (1160 − 800V ) dV
0.2
1.2
= ⎡⎣1160V − 400V 2 ⎤⎦
0.2
(
)
= 1160 × (1.2 − 0.2 ) − 400 1.22 − .22 kJ
= 1160 − 560kJ = 600kJ
0.2
− 85 = 215kJ / kg
1.5
1.2
u2 = 1.5 × 200 ×
− 85 = 155kJ / kg
1.5
∴ Δu = u2 − u1 = ( 275 − 215 ) = 40kJ / kg
u1 = 1.5 × 1000 ×
∴ ΔU = mΔu = 40 × 1.5 = 60kJ
∴ Q = ΔU + W = 60 + 600 = 660kJ
⇒ u = 1.5pv − 85kJ / kg
⎛ 1160 − 800v ⎞
= 1.5 ⎜
⎟ v − 85kJ / kg
1.5
⎝
⎠
2
= 1160v − 800v − 85kJ / kg
∂u
= 1160 − 1600v
∂v
∂u
1160
= 0∴ v =
= 0.725
∂v
1600
2
= 1160 × 0.725 − 800 × ( 0.725 ) − 85kJ / kg
for max imum u,
∴
umax .
U max
Q4.15
= 335.5kJ / kg
= 1.5umax = 503.25kJ
The heat capacity at constant pressure of a certain system is a function
of temperature only and may be expressed as
C p = 2.093 +
41.87
J/°C
t + 100
Where t is the temperature of the system in °C. The system is heated
while it is maintained at a pressure of 1 atmosphere until its volume
increases from 2000 cm3 to 2400 cm3 and its temperature increases from
0°C to 100°C.
(a) Find the magnitude of the heat interaction.
Page 37 of 265
First Law of Thermodynamics
Chapter 4
(b) How much does the internal energy of the system increase?
(Ans. (a) 238.32 J (b) 197.79 J)
373
Solution:
Q=
∫ C dT
t = T − 273
p
273
∴ t + 100 = T − 173
373
=
41.87 ⎞
⎛
∫ ⎜⎝ 2.093 + T − 173 ⎟⎠ dT
273
373
= ⎡⎣2.093T + 41.87 ln T − 173 ⎤⎦ 273
⎛ 200 ⎞
= 2.093 ( 373 − 273 ) + 41.87 ln ⎜
⎟
⎝ 100 ⎠
= 209.3 + 41.87 ln 2
= 238.32J
Q = ΔE + ∫ pdV
ΔE = Q − ∫ pdV
= Q − p ( V2 − V1 )
= 238.32 − 101.325 ( 0.0024 − 0.0020 ) × × 1000J
= ( 238.32 − 40.53 ) J
= 197.79J
Q4.16
Solution:
An imaginary engine receives heat and does work on a slowly moving
piston at such rates that the cycle of operation of 1 kg of working fluid
can be represented as a circle 10 cm in diameter on a p–v diagram on
which 1 cm = 300 kPa and 1 cm = 0.1 m3/kg.
(a) How much work is done by each kg of working fluid for each cycle of
operation?
(b) The thermal efficiency of an engine is defined as the ratio of work
done and heat input in a cycle. If the heat rejected by the engine in a
cycle is 1000 kJ per kg of working fluid, what would be its thermal
efficiency?
(Ans. (a) 2356.19 kJ/kg, (b) 0.702)
Given Diameter = 10 cm
π × 102
Work
∴ Area =
cm2 = 78.54 cm2
4
p
1 cm2 ≡ 300kPa × 0.1m3 / kg
= 30kJ
30 cm dia
∴ Total work done = 78.54 × 30kJ / kg
= 2356.2 kJ / kg
Heat rejected = 1000kJ
2356.2
× 100%
Therefore, η =
2356.2 + 1000
= 70.204%
Page 38 of 265
V
First Law of Thermodynamics
Chapter 4
Q4.17
Solution:
A gas undergoes a thermodynamic cycle consisting of three processes
beginning at an initial state where p1 = 1 bar, V1 = 1.5 m3 and U1 = 512 kJ.
The processes are as follows:
Compression with pV = constant to p2 = 2
(i) Process 1–2:
bar, U2 = 690 kJ
(ii) Process 2–3: W23 = 0, Q23 = –150 kJ, and
(iii) Process 3–1: W31 = +50 kJ. Neglecting KE and PE changes,
determine the heat interactions Q12 and Q31.
(Ans. 74 kJ, 22 kJ)
Q1−2 = ΔE + ∫ pdV
v2
Q1−2 = ( u2 − u1 ) + p1 V1 ∫
v1
dV
V
⎛p ⎞
= ( 690 − 512 ) + 100 × 1.5 × ln ⎜ 1 ⎟
⎝ p2 ⎠
= 178 − 103.972
= 74.03kJ
As W2-3 is ZERO so it is constant volume process. As W31 is +ive (positive) so
expansion is done.
∴ u3 = u2 − 150 = 540kJ
∴ Q31 = u1 − u3 + W
= ΔE + W = − ( 540 − 512 ) + 50
= − 28 + 50 = 22kJ
Q4.18
A gas undergoes a thermodynamic cycle consisting of the following
processes:
(i) Process 1–2: Constant pressure p = 1.4 bar, V1 = 0.028 m3, W12 = 10.5
kJ
(ii) Process 2–3: Compression with pV = constant, U3 = U2
(iii) Process 3–1: Constant volume, U1 – U3 = – 26.4 kJ. There are no
significant changes in KE and PE.
(a)
(b)
(c)
(d)
Sketch the cycle on a p–V diagram
Calculate the net work for the cycle in kJ
Calculate the heat transfer for process 1–2
Show that ∑ Q = ∑ W .
cycle
Solution:
( b ) W12
cycle
(Ans. (b) – 8.28 kJ, (c) 36.9 kJ)
= 10.5 kJ
Page 39 of 265
First Law of Thermodynamics
Chapter 4
(a)
3
W23 = ∫ pdV
3
2
3
dV
V
2
= p2 V2 ∫
pV = C
p
1.4 bar
⎛V ⎞
= p2 V2 ln ⎜ 3 ⎟
⎝ V2 ⎠
⎛V ⎞
= p2 V2 ln ⎜ 1 ⎟
⎝ V2 ⎠
u3
1
u1
2 u2
W12= 10.5kJ
0.028m3
V
⎛ 0.028 ⎞
= 1.4 × 100 × 0.103 × ln ⎜
⎟
⎝ 0.103 ⎠
⎡as
⎤
W 12 = p ( V2 − V1 )
⎢
⎥
10.5 = 1.4 × 100 ( V2 − 0.028 ) ⎥
= − 18.783kJ
⎢
⎢
⎥
V2 = 0.103 m3
⎢⎣∴
⎥⎦
W31 = 0 as constant volume
∴ Net work output = − 8.283 kJ
( c ) Q12
ans.(b)
= U 2 − U1 + W12
= 26.4 + 10.5kJ = 36.9kJ
(d) Q23 = U3 − U 2 + W23
= 0 − 18.783kJ = − 18.783 kJ
Q31 = U 2 − U3 + 0 = − 26.4kJ
∴
∑Q = Q
12
+ Q23 + Q31 = 36.9kJ − 18.783 − 26.4
= − 8.283kJ
∴
∑W = ∑Q
Pr oved.
Page 40 of 265
First Law Applied to Flow Process
Chapter 5
5. First Law Applied to Flow Process
Some Important Notes
• S.F.E.E. per unit mass basis
V 12
V 22
dQ
dW
h1 +
+ gZ1 +
= h2 +
+ gZ2 +
2
dm
2
dm
[h, W, Q should be in J/kg and C in m/s and g in m/s2]
V12
dQ
V22
dW
gZ1
gZ 2
h1 +
+
+
= h2 +
+
+
2000 1000 dm
2000 1000 dm
[h, W, Q should be in kJ/kg and C in m/s and g in m/s2]
• S.F.E.E. per unit time basis
⎛
⎞ dQ
V2
w1 ⎜ h1 + 1 + Z1 g ⎟ +
2
⎝
⎠ dτ
⎛
⎞ dWx
V2
= w2 ⎜ h2 + 2 + Z2 g ⎟ +
2
dτ
⎝
⎠
Where, w = mass flow rate (kg/s)
• Steady Flow Process Involving Two Fluid Streams at the Inlet
and Exit of the Control Volume
Mass balance
w
A 1V
v1
1
+
1
A 2V
v2
+ w
2
=
2
= w
A 3V
v3
3
3
+ w
+
Where, v = specific volume (m3/kg)
Page 41 of 265
4
A 4V
v4
4
First Law Applied to Flow Process
Chapter 5
Energy balance
⎛
⎞
⎛
⎞ dQ
V2
V2
w1 ⎜ h1 + 1 + Z1 g ⎟ + w2 ⎜ h2 + 2 + Z2 g ⎟ +
2
2
⎝
⎠
⎝
⎠ dτ
⎛
⎞
⎛
⎞ dWx
V32
V42
= w3 ⎜ h3 +
+ Z3 g ⎟ + w4 ⎜ h4 +
+ Z4 g ⎟ +
dτ
2
2
⎝
⎠
⎝
⎠
Questions with Solution P. K. Nag
Q5.1
A blower handles 1 kg/s of air at 20°C and consumes a power of 15 kW.
The inlet and outlet velocities of air are 100 m/s and 150 m/s respectively.
Find the exit air temperature, assuming adiabatic conditions. Take cp of
air is 1.005 kJ/kg-K.
(Ans. 28.38°C)
Solution:
2
1
t1 = 20°C
V1 = 100 m/s
1
2
V2 = 150 m/s
t2 = ?
dW
= – 15 kN
dt
From S.F.E.E.
⎛
⎛
V2
gZ1 ⎞ dQ
V2
gZ2 ⎞ dW
= w2 ⎜ h 2 + 2 +
w1 ⎜ h1 + 1 +
⎟+
⎟+
2000 1000 ⎠ dt
2000 1000 ⎠ dt
⎝
⎝
dQ
= 0.
Here w1 = w2 = 1 kg / s ; Z1 = Z2 ;
dt
1002
1502
∴
+ 0 = h2 +
− 15
h1 +
2000
2000
⎛
1002 1502 ⎞
∴
−
h2 − h1 = ⎜15 +
⎟
2000 2000 ⎠
⎝
or
or
Q5.2
Cp ( t2 − t1 ) = 8.75
t2 = 20 +
8.75
= 28.7°C
1.005
A turbine operates under steady flow conditions, receiving steam at the
following state: Pressure 1.2 MPa, temperature 188°C, enthalpy 2785
kJ/kg, velocity 33.3 m/s and elevation 3 m. The steam leaves the turbine
at the following state: Pressure 20 kPa, enthalpy 2512 kJ/kg, velocity 100
m/s, and elevation 0 m. Heat is lost to the surroundings at the rate of 0.29
kJ/s. If the rate of steam flow through the turbine is 0.42 kg/s, what is the
power output of the turbine in kW?
(Ans. 112.51 kW)
Page 42 of 265
First Law Applied to Flow Process
Chapter 5
Solution:
w1 = w2 = 0.42 kg / s
1
p1 = 1.2 MPa
t1 = 188°C
h1 = 2785 kJ/kg
V1 = 33.3 m/s
1
Z1 = 3 m
dQ
dt = – 0.29 kJ/s
dW
=?
dt
3m
2
By S.F.E.E.
2
p2 = 20 kPa
h2 = 2512 kJ/kg
V2 = 100 m/s
Z2 = 0
⎛
⎛
V2
g Z1 ⎞ dQ
V2
g Z2 ⎞ dW
w1 ⎜ h1 + 1 +
= w2 ⎜ h 2 + 2 +
⎟+
⎟+
2000 1000 ⎠ dt
2000 1000 ⎠ dt
⎝
⎝
⎧
⎧
⎫ dW
33.32 9.81 × 3 ⎫
1002
or
0.42 ⎨2785 +
+
+ 0⎬ +
⎬ − 0.29 = 0.42 ⎨2512 +
2000
1000 ⎭
2000
⎩
⎩
⎭ dt
Q5.3
Solution:
or
1169.655 = 1057.14 +
or
dW
= 112.515 kW
dt
dW
dt
A nozzle is a device for increasing the velocity of a steadily flowing
stream. At the inlet to a certain nozzle, the enthalpy of the fluid passing
is 3000 kJ/kg and the velocity is 60 m/s. At the discharge end, the
enthalpy is 2762 kJ/kg. The nozzle is horizontal and there is negligible
heat loss from it.
(a) Find the velocity at exists from the nozzle.
(b) If the inlet area is 0.1 m2 and the specific volume at inlet is 0.187
m3/kg, find the mass flow rate.
(c) If the specific volume at the nozzle exit is 0.498 m3/kg, find the exit
area of the nozzle.
(Ans. (a) 692.5 m/s, (b) 32.08 kg/s (c) 0.023 m2)
Find V2 i.e. Velocity at exit from S.F.E.E.
(a )
h1 +
V12
g Z1
V2
gZ2
dQ
dW
+
+
= h2 + 2 +
+
2000 1000 dm
2000 1000 dm
Data for a
1
h1 = 3000 kJ/kg
V1 = 60 m/s
2
h2 = 2762 kJ/kg
For data for c
3
2 v2 = 0.498 m /kg
Data for b
A1 = 0.1 m2 1
v1 = 0.187 m3/kg
Page 43 of 265
First Law Applied to Flow Process
Chapter 5
or
dQ
dW
= 0 and
=0
dm
dm
V2
V2
h1 + 1 = h2 + 2
2000
2000
2
2
V2 − V1
= ( h1 − h2 )
2000
V22 = V12 + 2000 ( h1 − h2 )
or
V2 =
Here Z1 = Z2 and
∴
or
602 + 2000 ( 3000 − 2762 )m / s
=
( b)
V12 + 2000 ( h1 − h 2 )
= 692.532 m / s
AV
Mass flow rate ( w ) = 1 1
v1
0.1 × 60
kg / s = 32.1kg / s
0.187
Mass flow rate is same so
=
(c )
A 2 × 692.532
0.498
A 2 = 8.023073 m2
32.0855613 =
or
Q5.4
In oil cooler, oil flows steadily through a bundle of metal tubes
submerged in a steady stream of cooling water. Under steady flow
conditions, the oil enters at 90°C and leaves at 30°C, while the water
enters at 25°C and leaves at 70°C. The enthalpy of oil at t°C is given by
h = 1.68 t + 10.5 × 10-4 t2 kJ/kg
Solution:
What is the cooling water flow required for cooling 2.78 kg/s of oil?
(Ans. 1.47 kg/s)
wo (h oi + 0 + 0) + wH2 O (h H2 Oi + 0 + 0) + 0 wo (h o,o + 0 + 0) + wH2 O (h H2 Oo + 0 + 0) + 0
Oil
1
90°C
Water
∴
25°C
1
wo (h oi − h o,o ) = wH2 0 (h H2 Oo − h H2 Oi )
2
30°C
70°C
2
hoi = 1.68 × 90 + 10.5 × 10–4 × 902 kJ/kg = 159.705 kJ/kg
ho,o = 1.68 × 30 + 10.5 × 10–4 × 362 kJ/kg = 51.395 kJ/kg
∴
Q5.5
2.78 × 108.36
kg/s
4.187 (70 − 25)
= 1.598815 kg/s 1.6 kg/s
WH2o
=
A thermoelectric generator consists of a series of semiconductor
elements (Figure) heated on one side and cooled on the other. Electric
current flow is produced as a result of energy transfer as heat. In a
Page 44 of 265
First Law Applied to Flow Process
Chapter 5
particular experiment the current was measured to be 0.5 amp and the
electrostatic potential at
(1) Was 0.8 volt above that at
(2) Energy transfer as heat to the hot side of the generator was taking
place at a rate of 5.5 watts. Determine the rate of energy transfer as
heat from the cold side and the energy conversion efficiency.
(Ans. Q2 = 5.1 watts, η = 0.073)
•
•
Q1 = E + Q2
Solution:
Q5.6
•
•
or
5.5 = 0.5 × 0.8 + Q2
or
Q2 = 5.1 watt
5.5 − 5.1
η=
× 100% = 7.273%
5.5
•
A turbo compressor delivers 2.33 m3/s at 0.276 MPa, 43°C which is heated
at this pressure to 430°C and finally expanded in a turbine which
delivers 1860 kW. During the expansion, there is a heat transfer of 0.09
MJ/s to the surroundings. Calculate the turbine exhaust temperature if
changes in kinetic and potential energy are negligible.
(Ans. 157°C)
Solution:
t = 93°C
C.C.
V1 = 2.33 m3/s ; p1 = 0.276 M Pa ; t = 930°C
1
1
dW
= 1860 kW
dt
2
2
dQ
= – 0.09 × 1000 kJ/s = – 90 kW
dt
dQ
dW
= w2 h 2 +
dt
dt
dW dQ
∴ w1 (h1 – h2) =
−
dt
dt
= 1860 – (–90) = 1950 kW
Page 45 of 265
w1 h1 +
or
First Law Applied to Flow Process
Chapter 5
p1 V1 =
∴
Or
∴
or
∴
Q5.7
m1R T1
p1V1
276 kPa × 2.33 m3 / s
= 7.091 kg/s
=
RT1
0.287 kJ/ kg × 316K
h1 – h2 = 275
Cp (t1 – t2) = 275
275
t1 – t2 =
273.60
1.005
t2 = 430 – 273.60
= 156.36º C
•
m1 =
A reciprocating air compressor takes in 2 m3/min at 0.11 MPa, 20°C
which it delivers at 1.5 MPa, 111°C to an aftercooler where the air is
cooled at constant pressure to 25°C. The power absorbed by the
compressor is 4.15 kW. Determine the heat transfer in
(a) The compressor
(b) The cooler
State your assumptions.
(Ans. – 0.17 kJ/s, – 3.76 kJ/s)
Solution:
(a)
∴
∴
dQ
dW
= w1 h 2 +
dt
dt
⎛ dQ ⎞
0.0436 (111.555 – 20.1) – 4.15 = ⎜
⎟
⎝ dt ⎠
w1 (h1 + 0 + 0) +
dQ
= –0.1622 kW
dt
•
V1 = 2 m3 /min
p1 = 0.11 MPa
t1 = 20°C
i.e. 1622 kW loss by compressor
dW
= – 4.15 kW
dt
1
2
t2 = 111°C
p2 = 1.5M Pa
2
1
3
Cooles
3
n
n
(p2 V2 - p1 V1 ) =
(mRT2 − mRT1 )
n -1
n −1
1.4
=
× 0.0436 × 0.287(111 − 20) kW
0.4
= 3.9854 kW
Compressor work =
dQ
= 3.9854 – 4.15 = –0.165 kW
dt
dQ
For cooler
dt
∴
(b)
Page 46 of 265
First Law Applied to Flow Process
Chapter 5
•
= m cP (t 2 − t1 )
= 0.0436 × 1.005 × (111 – 25) kJ/s
= 3.768348 kW to surroundings
Q5.8
Solution:
In water cooling tower air enters at a height of 1 m above the ground
level and leaves at a height of 7 m. The inlet and outlet velocities are 20
m/s and 30 m/s respectively. Water enters at a height of 8 m and leaves at
a height of 0.8 m. The velocity of water at entry and exit are 3 m/s and 1
m/s respectively. Water temperatures are 80°C and 50°C at the entry and
exit respectively. Air temperatures are 30°C and 70°C at the entry and
exit respectively. The cooling tower is well insulated and a fan of 2.25 kW
drives the air through the cooler. Find the amount of air per second
required for 1 kg/s of water flow. The values of cp of air and water are
1.005 and 4.187 kJ/kg K respectively.
(Ans. 3.16 kg/s)
a
Let air required is w1 kg/s
⎛ w
⎛
Va2
g Z1a ⎞
V1w
g Z1w ⎞ dQ
w
w
+
+
+
∴ w1a ⎜ h1a + 1 +
h
⎟+
⎟
1 ⎜ 1
2000 1000 ⎠
2000 1000 ⎠ dt
⎝
⎝
2
⎛ w
V2a 2
g Z2a ⎞
V2w
g Z2w ⎞ dW
a ⎛ a
w
+
+
= w2 ⎜ h2 +
⎟+
⎟ + w2 ⎜ h2 +
2000 1000 ⎠
2000 1000 ⎠
dt
⎝
⎝
dQ
∴ w1a = w2a = w (say) and
= 0 w1w = w2w = 1 kg/s
dt
2
a
w
V2 = 30 m/s
t2a = 70°C
7m
8m
a
w
V1 = 20 m/s
t1a = 30°C
1m
cap = 1.005 kJ/kg – K
w
V2 = 1 m/s, t2 = 50°C
w
w = w2w = 1 kg/s
0.8 m w 1
cp = 4.187 kJ/kg – K
dW
= – 2.25 kW
dt
⎧ a
⎫
V a − V2a
g
⎨(h1 − h2a ) + 1
+
( Z1a − Z2a ) ⎬
2000
1000
⎩
⎭
2
2
w
w
⎧
⎫ dW
V − V1
g
( Z1w − Z2w ) ⎬ +
= ⎨(h 2w − h1w ) + 2
+
2000
1000
⎩
⎭ dt
2
2
⎧
⎫
20 − 30
9.81
Or w ⎨1.005 × (30 − 70) +
+
(1 − 7) ⎬
2000
1000
⎩
⎭
Page 47 of 265
2
∴
w
V1 = 3 m/s, t1 = 80°C
2
First Law Applied to Flow Process
Chapter 5
= 4.187 (50 − 80) +
12 − 32 9.81
+
× (0.8 − 8) − 2.25
2000 1000
or – w × 40.509 = –127.9346
127.9346
∴
= 3.1582 kg/s ≈ 3.16 kg/s
w=
40.509
Q5.9
Air at 101.325 kPa, 20°C is taken into a gas turbine power plant at a
velocity of 140 m/s through an opening of 0.15 m2 cross-sectional area.
The air is compressed heated, expanded through a turbine, and
exhausted at 0.18 MPa, 150°C through an opening of 0.10 m2 crosssectional area. The power output is 375 kW. Calculate the net amount of
heat added to the air in kJ/kg. Assume that air obeys the law
pv = 0.287 (t + 273)
Where p is the pressure in kPa, v is the specific volume in m3/kg, and t is
the temperature in °C. Take cp = 1.005 kJ/kg K.
(Ans. 150.23 kJ/kg)
Solution:
•
Volume flow rate at inlet (V)1 = V1A1 m3/s = 21 m3/s
•
p V1
101.325 × 21
= 25.304 kg/s
Inlet mass flow rate ( w1 ) = 1
=
R T1
0.287 × 293
•
Volume flow rate at outlet = (V 2 ) =
=
1
w2 RT2
p2
25.304 × 0.287 × 423
= 17 m3/s
180
dW
= 375 kW
dt
CC
2
p1 = 101.325 kPa
1
t1 = 20°C
V1 = 140 m/s
A1 = 0.15 m2
•
2 p = 0.18 MPa = 180 kPa
2
t2 = 150°C
A2 = 0.1 m2
V2 = 171 m/s
V
17
Velocity at outlet = 2 =
= 170.66 m/s
A2
0.1
∴
Using S.F.E.E.
⎛
⎞ dQ
⎛
⎞ dW
V2
V22
w1 ⎜ h1 + 1 + 0 ⎟ +
= w2 ⎜ h 2 +
+ 0⎟ +
2000
2000
⎝
⎠ dt
⎝
⎠ dt
w1 = w 2 = w = 25.304 kg/s
∴
⎧
V 2 − V12 ⎫ dW
dQ
= w ⎨(h 2 − h1 ) + 2
⎬+
dt
2000 ⎭ dt
⎩
⎧
V 2 − V12 ⎫ dW
= w ⎨C p (t2 − t1 ) + 2
⎬+
2000 ⎭ dt
⎩
Page 48 of 265
First Law Applied to Flow Process
Chapter 5
⎧
1712 − 1402 ⎫
= 25.304 ⎨1.005 (150 − 20) +
⎬ + 375 kW
2000
⎩
⎭
= 3802.76 kW
dQ
dQ d t
=
w
dm
=
3802.76
= 150.28 kJ
kg
25.304
Q5.10
A gas flows steadily through a rotary compressor. The gas enters the
compressor at a temperature of 16°C, a pressure of 100 kPa, and an
enthalpy of 391.2 kJ/kg. The gas leaves the compressor at a temperature
of 245°C, a pressure of 0.6 MPa, and an enthalpy of 534.5 kJ/kg. There is
no heat transfer to or from the gas as it flows through the compressor.
(a) Evaluate the external work done per unit mass of gas assuming the
gas velocities at entry and exit to be negligible.
(b) Evaluate the external work done per unit mass of gas when the gas
velocity at entry is 80 m/s and that at exit is 160 m/s.
(Ans. 143.3 kJ/kg, 152.9 kJ/kg)
Solution:
(a)
V12
g Z1
V22
g Z2
dQ
dW
+
+
= h2 +
+
+
2000 1000 dm
2000 1000 dm
For V1 and V2 negligible and Z1 = Z 2 so
h1 +
dW
= h1 – h2 = (391.2 – 5345) kJ/kg
dm
= –143.3 kJ/kg i.e. work have to give
1
t1 = 16°C
p1 = 100 kPa
h1 = 391.2 kJ/kg
1
(b)
Q5.11
2
2
dQ
∴
=0
dt
t2 = 245°C
p2 = 0.6 mPa = 600 kPa
h2 = 534.5 kJ/kg
V1 = 80 m/s; V2 = 160 m/s
V 2 − V22
dW
So
= (h1 − h 2 ) + 1
dm
2000
2
80 − 1602
= –143.3 +
kJ/kg = (–143.3 – 9.6) kJ/kg
2000
= –152.9 kJ/kg i.e. work have to give
The steam supply to an engine comprises two streams which mix before
entering the engine. One stream is supplied at the rate of 0.01 kg/s with
an enthalpy of 2952 kJ/kg and a velocity of 20 m/s. The other stream is
supplied at the rate of 0.1 kg/s with an enthalpy of 2569 kJ/kg and a
velocity of 120 m/s. At the exit from the engine the fluid leaves as two
Page 49 of 265
First Law Applied to Flow Process
Chapter 5
Solution:
streams, one of water at the rate of 0.001 kg/s with an enthalpy of 420
kJ/kg and the other of steam; the fluid velocities at the exit are
negligible. The engine develops a shaft power of 25 kW. The heat
transfer is negligible. Evaluate the enthalpy of the second exit stream.
(Ans. 2402 kJ/kg)
dQ
∴
=0
dt
By mass balance
dW
= 25 kW
dt
w11 = 0.01 kg/s
h11 = 2952 kJ/kg
V11 = 20 m/s
h22 = ?
w22 = ?
V22 = 0
w12 = 0.1 kg/s
h12 = 2569 kJ/kg
V12 = 120 m/s
w21 = 0.001 kg/s
h21 = 420 kJ/kg
V21 = 0
W11 + W12 = W21 + W22
∴
W22 = 0.01 + 0.1 – 0.001 kg/s = 0.109 kg/s
⎛
⎛
V2 ⎞
V 2 ⎞ dQ
W11 ⎜ h11 + 11 ⎟ + W12 ⎜ h12 + 12 ⎟ +
∴
2000 ⎠
2000 ⎠ dt
⎝
⎝
dW
= W21 (h 21 ) + W22 × h 22 +
dt
2
2
⎛
⎞
⎛
20
120 ⎞
∴
0.01 ⎜ 2952 +
⎟ + 0.1 ⎜ 2569 +
⎟+0
2000 ⎠
2000 ⎠
⎝
⎝
= 0.001 × 420 + 0.109 × h22 + 25
or
29.522 + 257.62 = 0.42 + 0.109 × h22 + 25
or
286.722 = 0.109 × h22 + 25
or
h22 = 2401.2 kJ/kg
Q5.12
Solution:
The stream of air and gasoline vapour, in the ratio of 14: 1 by mass,
enters a gasoline engine at a temperature of 30°C and leaves as
combustion products at a temperature of 790°C. The engine has a
specific fuel consumption of 0.3 kg/kWh. The net heat transfer rate from
the fuel-air stream to the jacket cooling water and to the surroundings is
35 kW. The shaft power delivered by the engine is 26 kW. Compute the
increase in the specific enthalpy of the fuel air stream, assuming the
changes in kinetic energy and in elevation to be negligible.
(Ans. – 1877 kJ/kg mixture)
In 1 hr. this m/c will produce 26 kWh for that we need fuel
= 0.3 × 26 = 7.8 kg fuel/hr.
∴
Mass flow rate of fuel vapor and air mixture
Page 50 of 265
First Law Applied to Flow Process
Chapter 5
dW
= 26 kW
dt
wg=x kg/s
w1=15 x kg/s
o
t2=790 C
t1=30oC
wa=14 x kg/s
dQ
= – 35 kW
dt
w1 =
15 × 7.8
kg/s = 0.0325 kg/s
3600
Applying S.F.E.E.
dQ
dW
w1 h1 +
= w1 h 2 +
dt
dt
dQ dW
or
w1 (h2 – h1) =
−
dt
dt
dQ dW
−
dt
∴
h 2 – h1 = dt
w1
−35 − 26
= –1877 kJ/kg of mixture.
=
0.0325
An air turbine forms part of an aircraft refrigerating plant. Air at a
pressure of 295 kPa and a temperature of 58°C flows steadily into the
turbine with a velocity of 45 m/s. The air leaves the turbine at a pressure
of 115 kPa, a temperature of 2°C, and a velocity of 150 m/s. The shaft
work delivered by the turbine is 54 kJ/kg of air. Neglecting changes in
elevation, determine the magnitude and sign of the heat transfer per
unit mass of air flowing. For air, take cp = 1.005 kJ/kg K and the enthalpy
h = cp t.
(Ans. + 7.96 kJ/kg)
Q5.13
Solution:
V12
V22
dW
dQ
+
+
+
h
= 2
2000 dm
2000 dm
2
2
dQ (h − h ) + V2 − V1 + dW
or
=
2
1
2000
dm
dm
dW
= 54 kJ/kg
dm
h1 +
2
2
150 − 45
+ 54 kJ/kg
2000
= –56.28 + 10.2375 + 54 kJ/kg
= 7.9575 kJ/kg (have to give to the
system)
= (2.01 − 58.29) +
Q5.14
p1 = 295 kPa
t1 = 58°C
V1 = 45 m/s
1
h1 = CPt
1
= 1.005 × 58
= 58.29 kJ/kg
2 p2 = 115 kPa
t2 = 2°C
z 1 = z2
V2 = 150 m/s
2 h2 = 2.01 kJ/kg
In a turbo machine handling an incompressible fluid with a density of
1000 kg/m3 the conditions of the fluid at the rotor entry and exit are as
given below:
Exit
Inlet
Pressure
1.15 MPa
0.05 MPa
Velocity
30 m/s
Page 51 of 265 15.5 m/s
First Law Applied to Flow Process
Chapter 5
Height above datum
10 m
2m
If the volume flow rate of the fluid is 40 m3/s, estimate the net energy
transfer from the fluid as work.
(Ans. 60.3 MW)
Solution:
By S.F.E.E.
⎛p
⎞ dQ
⎛p
⎞ dW
V2
V2
w ⎜ 1 + 1 + g Z1 ⎟ +
= w ⎜ 2 + 2 + g Z2 ⎟ +
2
2
dt
⎝ρ
⎠ dt
⎝ ρ
⎠
1
p1 = 1.15 M Pa
= 1150 kPa
V1 = 30 m/s
z1 = 10 m
1
p2 = 0.05MPa
= 50 kPa
V2 = 15.5 m/s
z2 = 2 m
2
2
datum
Flow rate = 40 m3/s ≡ 40 × 1000 kg/s = w (say)
∴
Or
⎛ 1150
302
9.81 × 10 ⎞
40000 ⎜
+
+
⎟+0
1000 ⎠
⎝ 1000 2000
⎧ p − p2 V12 − V22
⎫
dW
= 40000 ⎨ 1
+
+ g( Z1 − Z2 ) ⎬
2
dt
⎩ ρ
⎭
2
2
⎧1150 − 50 30 − 15.5
9.81 × (10 − 2) ⎫
+
+
= 40000 ⎨
⎬ kW
2000
1000
⎩ 1000
⎭
= 60.3342 MW
Q5.15
Solution:
A room for four persons has two fans, each consuming 0.18 kW power,
and three 100 W lamps. Ventilation air at the rate of 80 kg/h enters with
an enthalpy of 84 kJ/kg and leaves with an enthalpy of 59 kJ/kg. If each
person puts out heat at the rate of 630 kJ/h determine the rate at which
heat is to be removed by a room cooler, so that a steady state is
maintained in the room.
(Ans. 1.92 kW)
dQperson
4 × 630
kJ/s = 0.7 kW
= +
dt
3600
dQelectic
3 × 100
kW = 0.66 kW
= + 0.18 × 2 +
dt
1000
dQ
= 1.36 kW
∴
dt
Page 52 of 265
First Law Applied to Flow Process
Chapter 5
Electric
Man
w1 = 80 kg/hr
1
kg/s
45
h1 = 84 kJ/kg
w 2 = 1 kg/s
=
45
h2 = 59 kJ/kg
For steady state
dQ
dW
w1 h1 +
= w2 h 2 +
dt
dt
dW
dQ
1
× (84 − 59) + 1.36 kW
= w1 h1 − w2 h 2 +
=
∴
dt
45
dt
= 1.9156 kW
Q5.16
Air flows steadily at the rate of 0.4 kg/s through an air compressor,
entering at 6 m/s with a pressure of 1 bar and a specific volume of 0.85
m3/kg, and leaving at 4.5 m/s with a pressure of 6.9 bar and a specific
volume of 0.16 m3/kg. The internal energy of the air leaving is 88 kJ/kg
greater than that of the air entering. Cooling water in a jacket
surrounding the cylinder absorbs heat from the air at the rate of 59 W.
Calculate the power required to drive the compressor and the inlet and
outlet cross-sectional areas.
(Ans. 45.4 kW, 0.057 m2, 0.0142 m2)
Solution:
By S.F.E.E.
⎛
⎞ dQ
⎛
⎞ dW
V2
V22
= w2 ⎜ u2 + p2 v 2 +
+ 0⎟ +
w1 ⎜ u1 + p1 v1 + 1 + 0 ⎟ +
2000
2000
⎝
⎠ dt
⎝
⎠ dt
2
2
⎡
V − V2 ⎤ dQ
dW
Or
= ⎢(u1 − u2 ) + (p1 v1 - p2 v 2 ) + 1
⎥+
dt
2000 ⎦ dt
⎣
= 0.4 [– 88 + 85 – 110.4 + 0.0076] – 0.059
= – 45.357 – 0.059
= – 45.416 kW [have to give to compressor]
dQ
= – 59 W
dt
w1 = 0.4 kg/s
V1 = 6 m/s
p1 = 1 bar = 100 kPa
v1 = 0.85 m3/kg
u1 = ?
w1 =
w2 =
A 2 V2
v2
A1 V1
v1
∴ A2 =
1
2
1
2
∴ A1 =
w2 = 0.4 kg/s = W1
V2 = 4.5 m/s
p2 = 6.9 bar = 690 kPa
v2 = 0.16 m3/kg
u2 = u1 + 88 kJ/kg
w1 v1
0.4 × 0.85
= 0.0567 m2
=
6
V1
w2 v 2
0.4 × 0.16
= 0.01422 m2
=
4.5
V2
Page 53 of 265
Page 54 of 265
Second Law of Thermodynamics
Chapter 6
6.
Second Law of Thermodynamics
Some Important Notes
Regarding Heat Transfer and Work Transfer
•
Heat transfer and work transfer are the energy interactions.
•
Both heat transfer and work transfer are boundary phenomena.
•
It is wrong to say 'total heat' or 'heat content' of a closed system, because heat or work is
not a property of the system.
•
Both heat and work are path functions and inexact differentials.
•
Work is said to be a high grade energy and heat is low grade energy.
•
HEAT and WORK are NOT properties because they depend on the path and end states.
•
HEAT and WORK are not properties because their net change in a cycle is not zero.
•
Clausius' Theorem: The cyclic integral of d Q/T for a reversible cycle is equal to zero.
or
v∫
R
dQ
=0
T
•
The more effective way to increase the cycle efficiency is to decrease T2.
•
Comparison of heat engine, heat pump and refrigerating machine
QC
T
= C
QH
TH
hence, ηCarnot,HE = 1 −
QC
T
= 1− C
QH
TH
Page 55 of 265
Second Law of Thermodynamics
Chapter 6
COPCarnot ,HP =
COPCarnot ,R =
QH
QH
TH
=
=
Wcycle QH − QC TH − TC
QC
QC
TC
=
=
Wcycle QH − QC TH − TC
Questions with Solution P. K. Nag
Q6.1
Solution:
An inventor claims to have developed an engine that takes in 105 MJ at a
temperature of 400 K, rejects 42 MJ at a temperature of 200 K, and
delivers 15 kWh of mechanical work. Would you advise investing money
to put this engine in the market?
(Ans. No)
Maximum thermal efficiency of his engine possible
200
ηm a x = 1 −
= 50%
400
∴
That engine and deliver output = η × input
= 0.5 × 105 MJ
= 52.5 MJ = 14.58 kWh
As he claims that his engine can deliver more work than ideally possible so I
would not advise to investing money.
Q6.2
Solution:
If a refrigerator is used for heating purposes in winter so that the
atmosphere becomes the cold body and the room to be heated becomes
the hot body, how much heat would be available for heating for each kW
input to the driving motor? The COP of the refrigerator is 5, and the
electromechanical efficiency of the motor is 90%. How does this compare
with resistance heating?
(Ans. 5.4 kW)
COP =
desired effect
input
(COP) ref. = (COP) H.P – 1
H
or
6=
W
H
So input (W) =
6
But motor efficiency 90% so
Electrical energy require (E) =
∴ (COP) H.P. = 6
W
H
=
0.9
0.9 × 6
= 0.1852 H
= 18.52% of Heat (direct heating)
100
kW
H=
= 5.3995 kW
18.52 kW of work
Q6.3
Using an engine of 30% thermal efficiency to drive a refrigerator having
a COP of 5, what is the heat input into the engine for each MJ removed
from the cold body by the refrigerator?
(Ans. 666.67 kJ)
Page 56 of 265
Second Law of Thermodynamics
Chapter 6
Solution:
If this system is used as a heat pump, how many MJ of heat would be
available for heating for each MJ of heat input to the engine?
(Ans. 1.8 MJ)
COP of the Ref. is 5
So for each MJ removed from the cold body we need work
1MJ
= 200 kJ
5
For 200 kJ work output of heat engine hair η = 30%
200 kJ
= 666.67 kJ
We have to supply heat =
0.3
Now
COP of H.P. = COP of Ref. + 1
=5+1=6
Heat input to the H.E. = 1 MJ
∴
Work output (W) = 1 × 0.3 MJ = 300 kJ
That will be the input to H.P.
Q
∴ ( COP ) H.P = 1
W
∴
Q1 = (COP) H.P. × W = 6 × 300 kJ = 1.8 MJ
=
Q6.4
An electric storage battery which can exchange heat only with a
constant temperature atmosphere goes through a complete cycle of two
processes. In process 1–2, 2.8 kWh of electrical work flow into the battery
while 732 kJ of heat flow out to the atmosphere. During process 2–1, 2.4
kWh of work flow out of the battery.
(a) Find the heat transfer in process 2–1.
(b) If the process 1–2 has occurred as above, does the first law or the
second law limit the maximum possible work of process 2–1? What is
the maximum possible work?
(c) If the maximum possible work were obtained in process 2–1, what
will be the heat transfer in the process?
(Ans. (a) – 708 kJ (b) Second law, W2–1 = 9348 kJ (c) Q2–1 = 0)
Solution:
From the first Law of thermodynamics
(a) For process 1–2
Q1–2 = E2 – E1 + W1–2
–732 = (E2 – E1) – 10080
[2.8 kWh = 2.8 × 3600 kJ ]
–W
–Q
∴ E2 – E1 = 9348 kJ
For process 2–1
Q 21 = E1 – E 2 + W21
+W
= –9348 + 8640
= –708 kJ i.e. Heat flow out to the atmosphere.
(b) Yes Second Law limits the maximum possible work. As Electric energy stored
in a battery is High grade energy so it can be completely converted to the
work. Then,
W = 9348 kJ
Page 57 of 265
Second Law of Thermodynamics
Chapter 6
(c)
Q6.5
Solution:
Q21 = –9348 + 9348 = 0 kJ
A household refrigerator is maintained at a temperature of 2°C. Every
time the door is opened, warm material is placed inside, introducing an
average of 420 kJ, but making only a small change in the temperature of
the refrigerator. The door is opened 20 times a day, and the refrigerator
operates at 15% of the ideal COP. The cost of work is Rs. 2.50 per kWh.
What is the monthly bill for this refrigerator? The atmosphere is at 30°C.
(Ans. Rs. 118.80)
275
275
Ideal COP of Ref. =
=
= 9.82143
30 − 2
28
Actual COP = 0.15 × COP ideal = 1.4732
303 K
Heat to be removed in a day
Q1 = Q2 + W
(Q2) = 420 × 20 kJ
W
= 8400 kJ
R
∴
Q2
Work required = 5701.873 kJ/day
= 1.58385 kWh/day
275 K
Electric bill per month = 1.58385 × 0.32 × 30 Rupees
= Rs. 15.20
Q6.6
A heat pump working on the Carnot cycle takes in heat from a reservoir
at 5°C and delivers heat to a reservoir at 60°C. The heat pump is driven
by a reversible heat engine which takes in heat from a reservoir at 840°C
and rejects heat to a reservoir at 60°C. The reversible heat engine also
drives a machine that absorbs 30 kW. If the heat pump extracts 17 kJ/s
from the 5°C reservoir, determine
(a) The rate of heat supply from the 840°C source
(b) The rate of heat rejection to the 60°C sink.
(Ans. (a) 47.61 kW; (b) 34.61 kW)
Solution:
COP of H.P.
∴
∴
333
= 6.05454
=
333 − 278
Q3 = WH.P. + 17
WH.P. + 17
= 6.05454
WH.P.
17
= 5.05454
WH.P.
17
= 3.36 kW
5.05454
∴ Work output of the Heat engine
WH.E. = 30 + 3.36 = 33.36 kW
333
η of the H.E. = 1 −
= 0.7
1113
∴
WH.P. =
Page 58 of 265
278 K
1113 K
17 kW
WHP
H.P.
W
Q3
30 kW
333 K
Q1
H.E.
Q2
Second Law of Thermodynamics
Chapter 6
(a) ∴
∴
W
= 0.7
Q1
W
Q1 =
= 47.61 kW
0.7
(b) Rate of heat rejection to the 333 K
(i) From H.E. = Q1 – W = 47.61 – 33.36 = 14.25
kW
(ii) For H.P. = 17 + 3.36 = 20.36 kW
∴ Total = 34.61 kW
Q6.7
Solution:
A refrigeration plant for a food store operates as a reversed Carnot heat
engine cycle. The store is to be maintained at a temperature of – 5°C and
the heat transfer from the store to the cycle is at the rate of 5 kW. If heat
is transferred from the cycle to the atmosphere at a temperature of 25°C,
calculate the power required to drive the plant.
(Ans. 0.56 kW)
= 8.933
( COP ) R = 298268
– 268
298 K
5 kW
=
W
∴
Q6.8
Solution:
W=
W
5
kW = 0.56 kW
8.933
Q2 = (5 +W)kW
R
Q1 = 5 kW
268 K
A heat engine is used to drive a heat pump. The heat transfers from the
heat engine and from the heat pump are used to heat the water
circulating through the radiators of a building. The efficiency of the
heat engine is 27% and the COP of the heat pump is 4. Evaluate the ratio
of the heat transfer to the circulating water to the heat transfer to the
heat engine.
(Ans. 1.81)
For H.E.
1−
Q2
= 0.27
Q1
Page 59 of 265
Second Law of Thermodynamics
Chapter 6
Q2
= 0.73
Q1
Q2 = 0.73 Q1
T1
Q1
H.E.
W = Q1 – Q2 = 0.27 Q1
Q2
For H.P.
Q4
=4
W
∴
Q4 = 4W = 1.08 Q1
T2
Q4
H.P.
∴ Q2 + Q4 = (0.73 + 1.08) Q1 = 1.81 Q1
∴
W
Heat transfer to the circulating water
Heat for to the Heat Engine
W
Q3
T3
1.81 Q1
= 1.81
Q1
If 20 kJ are added to a Carnot cycle at a temperature of 100°C and 14.6
kJ are rejected at 0°C, determine the location of absolute zero on the
Celsius scale.
(Ans. – 270.37°C)
Q1
φ(t1 )
=
Let φ (t) = at + b
Q2
φ(t 2 )
Q1
at + b
= 1
∴
Q2
at 2 + b
=
Q6.9
Solution:
or
∴
20
a × 100 + b
a
=
= × 100 + 1
14.6
b
a×0+ b
a
= 3.6986 × 10–3
b
For absolute zero, Q2 = 0
Q1
a ×100 + b
=
0
a×t+b
or
a×t+b=0
−b
1
t=
or
= −
= –270.37º C
a
3.6986 × 10 −3
Two reversible heat engines A and B are arranged in series, A rejecting
heat directly to B. Engine A receives 200 kJ at a temperature of 421°C
from a hot source, while engine B is in communication with a cold sink
at a temperature of 4.4°C. If the work output of A is twice that of B, find
(a) The intermediate temperature between A and B
(b) The efficiency of each engine
(c) The heat rejected to the cold sink
(Ans. 143.4°C, 40% and 33.5%, 80 kJ)
∴
Q6.10
Page 60 of 265
Second Law of Thermodynamics
Chapter 6
Solution:
Q3
Q 2 − Q3
Q1
Q
Q − Q2
=
=
= 2 = 1
277.4
694
T
694 − T
T − 277.4
Hence Q1 – Q2 = 2 W2
Q2 – Q3 = W2
2
1
=
∴
694 − T
T − 277.4
or
or
2T – 277.4 × 2 = 694 – T
T = 416.27 K = 143.27º C
(b)
η1 = 40%
Q6.11
Solution:
Q1
HE
Q2
Q2
HE
277.4
= 33.36%
416.27
416.27
× 200 kJ = 119.96 kJ ;
Q2 =
694
277.4
× 119.96 = 79.94 kJ
Q1 =
416.27
η2 = 1 −
(c)
694 K
W1
T3
W2
Q3
277.4 K
A heat engine operates between the maximum and minimum
temperatures of 671°C and 60°C respectively, with an efficiency of 50% of
the appropriate Carnot efficiency. It drives a heat pump which uses
river water at 4.4°C to heat a block of flats in which the temperature is
to be maintained at 21.1°C. Assuming that a temperature difference of
11.1°C exists between the working fluid and the river water, on the one
hand, and the required room temperature on the other, and assuming
the heat pump to operate on the reversed Carnot cycle, but with a COP
of 50% of the ideal COP, find the heat input to the engine per unit heat
output from the heat pump. Why is direct heating thermodynamically
more wasteful?
(Ans. 0.79 kJ/kJ heat input)
273 + 60
333
Carnot efficiency (η) = 1 −
= 1−
= 0.64725
273 + 671
944
Actual (η) = 0.323623 = 1 −
Q1′
Q1
Page 61 of 265
Second Law of Thermodynamics
Chapter 6
Q1′
= 0.6764
Q1
Ideal COP
305.2
= 7.866
=
305.2 – 266.4
Actual COP
Q
= 3.923 = 3
W
if Q3 = 1 kJ
Q3
1
=
∴
W=
3.923
3.923
∴
block.
Q6.12
Solution:
= 0.2549 kJ/kJ heat input to block
W = Q1 − Q1′ = Q1 – 0.6764 Q1 =
0.2549
0.2549
Q1 =
(1 − 0.6764)
= 0.7877 kJ/kJ heat input to
An ice-making plant produces ice at atmospheric pressure and at 0°C
from water. The mean temperature of the cooling water circulating
through the condenser of the refrigerating machine is 18°C. Evaluate the
minimum electrical work in kWh required to produce 1 tonne of ice (The
enthalpy of fusion of ice at atmospheric pressure is 333.5 kJ/kg).
(Ans. 6.11 kWh)
273
= 15.2
Maximum (COP) =
291 − 273
18°C
Q
∴
= 15.2
291
K
W
min
or
Wmin =
Q
1000 × 333.5
kJ
=
15.2
15.2
= 21.989 MJ = 6.108 kWh
Q6.13
W
Q2
R
Q1
273 K
0°C
A reversible engine works between three thermal reservoirs, A, B and C.
The engine absorbs an equal amount of heat from the thermal reservoirs
A and B kept at temperatures TA and TB respectively, and rejects heat to
the thermal reservoir C kept at temperature TC. The efficiency of the
engine is α times the efficiency of the reversible engine, which works
between the two reservoirs A and C. prove that
TA
T
= (2α - 1) + 2(1 - α ) A
TB
TC
Page 62 of 265
Second Law of Thermodynamics
Chapter 6
Solution:
η of H.E. between A and C
T ⎞
⎛
η A = ⎜1 − C ⎟
TA ⎠
⎝
T ⎞
⎛
η of our engine = α ⎜1 − C ⎟
TA ⎠
⎝
Q
Q
Here Q 2 = 1 × TC = Q3 = 1 × TC
TA
TB
∴ Total Heat rejection
1 ⎞
⎛ 1
(Q2 + Q3) = Q1TC ⎜
+
⎟
⎝ TA TB ⎠
Total Heat input = 2Q1
A
B
TA
TB
Q1
Q1
H.E.
H.E.
Q2
Q3
TC
C
⎡
⎛ 1
1 ⎞⎤
+
⎢ Q1Tc ⎜
⎟⎥
T
T
A
B
⎝
⎠⎥
η of engine = ⎢1 −
⎢⎣
⎥⎦
2Q1
α TC
T
T
= 1− C − C
TA
2 TA 2 TB
Multiply both side by TA and divide by TC
T
T
1 1 TA
α A − α= A − −
or
TC
TC 2 2 TB
TA
T
= (2α − 1) + 2(1 − α ) A Proved
or
TB
TC
∴
α−
Q6.14
A reversible engine operates between temperatures T1 and T (T1 > T).
The energy rejected from this engine is received by a second reversible
engine at the same temperature T. The second engine rejects energy at
temperature T2 (T2 < T).
Show that:
(a) Temperature T is the arithmetic mean of temperatures T1 and T2 if
the engines produce the same amount of work output
(b) Temperature T is the geometric mean of temperatures T1 and T2 if
the engines have the same cycle efficiencies.
Solution:
(a) If they produce same Amount and work
Then W1 = W2
or η1Q1 = η 2 Q 2
T ⎞
T ⎞⎛T ⎞
⎛
⎛
or ⎜1 − ⎟ ⎜ 1 ⎟ Q2 = ⎜1 − 2 ⎟ Q2
T1 ⎠ ⎝ T ⎠
⎝
T⎠
⎝
Q
Q
We know that 1 = 2
T
T1
T
or
Q1 = 1 Q2
T
Page 63 of 265
Second Law of Thermodynamics
Chapter 6
T1
T
−1 = 1 − 2
T
T
T1 + T2
=2
or
T
T + T2
or
T= 1
2
i.e., Arithmetic mean and T1, T2
or
(b) If their efficiency is same then
T
T
1−
= 1− 2
T
T1
or
∴
Q6.15
Solution:
T=
T1T2
(as T is + ve so –ve sign neglected)
T is Geometric mean of T1 and T2.
T1
Q1
H.E.
W1
Q2
T
Q2
H.E.
W2
Q3
T2
Two Carnot engines A and B are connected in series between two
thermal reservoirs maintained at 1000 K and 100 K respectively. Engine
A receives 1680 kJ of heat from the high-temperature reservoir and
rejects heat to the Carnot engine B. Engine B takes in heat rejected by
engine A and rejects heat to the low-temperature reservoir. If engines A
and B have equal thermal efficiencies, determine
(a) The heat rejected by engine B
(b) The temperature at which heat is rejected by engine, A
(c) The work done during the process by engines, A and B respectively.
If engines A and B deliver equal work, determine
(d) The amount of heat taken in by engine B
(e) The efficiencies of engines A and B
(Ans. (a) 168 kJ, (b) 316.2 K, (c) 1148.7, 363.3 kJ,
(d) 924 kJ, (e) 45%, 81.8%)
As their efficiency is same so
ηA = ηB
T
100
= 1−
or 1 −
1000
T
(b) T = 1000 × 100 = 316.3K
Q2 =
Q1
1680 × 316.3
×T =
1000
1000
= 531.26 kJ
Q2
531.26 × 100
× 100 =
316.3
316.3
= 168 kJ as (a)
(c) WA = Q1 – Q2 = (1880 – 531.26) kJ
= 1148.74 kJ
WB = (531.26 – 168) kJ
= 363.26 kJ
(a) Q3 =
Page 64 of 265
Second Law of Thermodynamics
Chapter 6
100 + 1000
= 550 K
2
Q1
1680 × 550
×T =
= 924 kJ
∴ Q2 =
1000
1000
550
= 0.45
(e) η A = 1 −
1000
100
= 0.8182
ηB = 1 −
550
(d) If the equal work then T =
Q6.16
Solution :
A heat pump is to be used to heat a house in winter and then reversed to
cool the house in summer. The interior temperature is to be maintained
at 20°C. Heat transfer through the walls and roof is estimated to be 0.525
kJ/s per degree temperature difference between the inside and outside.
(a) If the outside temperature in winter is 5°C, what is the minimum
power required to drive the heat pump?
(b) If the power output is the same as in part (a), what is the maximum
outer temperature for which the inside can be maintained at 20°C?
(Ans. (a) 403 W, (b) 35.4°C)
(a) Estimated Heat rate
293 K
= 0.525 × (20 – 5) kJ/s = 7.875 kJ/s
20°C
•
293
Q = 7875 kJ/s
COP =
= 19.53
•
293 − 278
W
H.P.
•
Q
Wmin =
(COP)max
5°C
278 K
7.875
= 0.403 kW = 403 W
=
Winter
19.53
•
(b) Given W = 403 W
•
Heat rate (Q1 ) = 0.525 (T – 293) kW
= 525(T – 293) W
∴
525(T − 293)
293
COP =
=
403
(T − 293)
403 × 293
= 15
525
or
T = 308 K = 35º C
∴ Maximum outside Temperature = 35ºC
or (T – 293) =
Q6.17
T
R
•
Q1
293 K
Consider an engine in outer space which operates on the Carnot cycle.
The only way in which heat can be transferred from the engine is by
radiation. The rate at which heat is radiated is proportional to the
fourth power of the absolute temperature and to the area of the
radiating surface. Show that for a given power output and a given T1, the
area of the radiator will be a minimum when
T2 3
=
T1 4
Page 65 of 265
Second Law of Thermodynamics
Chapter 6
Solution:
Heat have to radiate = Q2
∴
Q2 = σ AT24
T1
From engine side
∴
∴
or
A=
or
Q6.18
Solution:
H.E.
W
Q2
T2
W⎧
1
W ⎧ T2 ⎫
⎫
⎬ =
⎨
⎬
3
4 ⎨
σ ⎩ T1T2 − T24 ⎭
σ T2 ⎩ T1 − T2 ⎭
For minimum Area
∂A
=0
or
∂ T2
or
or
Q1
Q1
Q
W
= 2 =
T1
T2
T1 − T2
WT2
Q2 =
T1 − T2
WT2
= σ AT24
T1 − T2
∂
{T1T23 − T24 } = 0
∂ T2
T1 × 3 T22 − 4 T23 = 0
3T1 = 4T2
T2
3
=
proved
4
T1
It takes 10 kW to keep the interior of a certain house at 20°C when the
outside temperature is 0°C. This heat flow is usually obtained directly by
burning gas or oil. Calculate the power required if the 10 kW heat flow
were supplied by operating a reversible engine with the house as the
upper reservoir and the outside surroundings as the lower reservoir, so
that the power were used only to perform work needed to operate the
engine.
(Ans. 0.683 kW)
COP of the H.P.
20°C
203 K
10
293
=
W
293 − 273
10 kW
or
W=
10 × 20
kW
293
W
H.P.
= 683 W only.
273 K
Q6.19
Solution:
Prove that the COP of a reversible refrigerator operating between two
given temperatures is the maximum.
Suppose A is any refrigerator and B is reversible refrigerator and also assume
(COP)A > (COP) B
and
Q1A = Q1B = Q
Page 66 of 265
Second Law of Thermodynamics
Chapter 6
or
or
or
Q1 A Q1B
>
WA
WB
Q
Q
>
WA WB
WA < WB
T1
Q2A
WA
Q2B
WB
R1
Q1A
Then
we
reversed
the
reversible refrigerator ‘B’ and
then
work
output
of
refrigerator ‘B’ is WB and heat
rejection is Q1B = Q (same)
So we can directly use Q to
feed for refrigerator and
Reservoir ‘T2’ is eliminated
then also a net work output
(WB – WA) will be available.
But it violates the KelvinPlank statement i.e. violates
Second Law of thermodynamic
so our assumption is wrong.
So (COP) R ≥ (COP) A
Q6.20
Solution:
Q6.21
Solution:
R2
Q1B
T2
T1 > T2 and T1 and T2 fixed
T1
WA
R
H.E.
Q1A
WB
Q1B
T2
A house is to be maintained at a temperature of 20°C by means of a heat
pump pumping heat from the atmosphere. Heat losses through the walls
of the house are estimated at 0.65 kW per unit of temperature difference
between the inside of the house and the atmosphere.
(a) If the atmospheric temperature is – 10°C, what is the minimum
power required driving the pump?
(b) It is proposed to use the same heat pump to cool the house in
summer. For the same room temperature, the same heat loss rate,
and the same power input to the pump, what is the maximum
permissible atmospheric temperature?
(Ans. 2 kW, 50°C)
Same as 6.16
A solar-powered heat pump receives heat from a solar collector at Th,
rejects heat to the atmosphere at Ta, and pumps heat from a cold space
at Tc. The three heat transfer rates are Qh, Qa, and Qc respectively.
Derive an expression for the minimum ratio Qh/Qc, in terms of the three
temperatures.
If Th = 400 K, Ta = 300 K, Tc = 200 K, Qc = 12 kW, what is the minimum Qh?
If the collector captures 0.2 kW 1 m2, what is the minimum collector area
required?
(Ans. 26.25 kW, 131.25 m2)
Q
Woutput = h (Th − Ta )
Th
Page 67 of 265
Second Law of Thermodynamics
Chapter 6
Winput =
Qc
(Ta − Tc )
Tc
As they same
So
Th
(T - T )
T
Qh
= h × a c
Tc (Th - Ta )
Qc
Qh
W
H.E.
400 ⎧ 300 − 200 ⎫
×⎨
⎬ kW
200 ⎩ 400 − 300 ⎭
= 24 kW
2.4
= 120 m2
Required Area (A) =
0.2
Qh = 12 ×
Ta
Qa atm.
R
Qc
Tc
Q6.22
A heat engine operating between two reservoirs at 1000 K and 300 K is
used to drive a heat pump which extracts heat from the reservoir at 300
K at a rate twice that at which the engine rejects heat to it. If the
efficiency of the engine is 40% of the maximum possible and the COP of
the heat pump is 50% of the maximum possible, what is the temperature
of the reservoir to which the heat pump rejects heat? What is the rate of
heat rejection from the heat pump if the rate of heat supply to the
engine is 50 kW?
(Ans. 326.5 K, 86 kW)
ηactual = 0.4 ⎛⎜1 −
Solution:
300 ⎞
⎟ = 0.28
1000 ⎠
⎝
W = 0.28 Q1
Q2 = Q1 – W = 0.72 Q1
Q3 = 2 Q2 + W = 1.72 Q1
∴
1.72 Q1
0.28 Q1
T
× (0.5)
=
T − 300
6.143 T – 300 × 6.143 = T × 0.5
T = 326.58 K
Q3 = 1.72 × 50 kW = 86 kW
∴ (COP)actual =
or
or
1000 K
TK
Q3 = 2Q2 +W
Q1
H.E.
W
Q2
H.P.
2Q2
300 K
Q6.23
A reversible power cycle is used to drive a reversible heat pump cycle.
The power cycle takes in Q1 heat units at T1 and rejects Q2 at T2. The
heat pump abstracts Q4 from the sink at T4 and discharges Q3 at T3.
Develop an expression for the ratio Q4/Q1 in terms of the four
temperatures.
Q4 T4 (T1 − T2 ) ⎞
⎛
⎜ Ans. Q = T (T − T ) ⎟
1
1
3
4 ⎠
⎝
Solution:
For H.E.
Page 68 of 265
Second Law of Thermodynamics
Chapter 6
Work output (W) =
For H.P.
Work input (W) =
∴
Q1
(T1 − T2 )
T1
Q4
(T3 − T4 )
T4
Q1
Q
(T1 − T2 ) = 4 (T3 − T4 )
T1
T4
T1
T3
Q3
Q1
H.E.
W
Q2
T3
H.P.
Q4
T4
Q4 T4 ⎧ T1 − T2 ⎫
=
⎨
⎬
Q1 T1 ⎩ T3 − T4 ⎭
This is the expression.
or
Q6.24
Prove that the following propositions are logically equivalent:
(a) A PMM2 is Impossible
(b) A weight sliding at constant velocity down a frictional inclined plane
executes an irreversible process.
Solution:
Applying First Law of Thermodynamics
Q12 = E2 – E1 + W1.2
or
0 = E2 – E1 – mgh
or
E1 – E2 = mgh
H
Page 69 of 265
Entropy
Chapter 7
7.
Entropy
Some Important Notes
1.
Clausius theorem:
2.
Sf – Si =
f
∫
i
⎛ dQ ⎞
=0
⎟
T ⎠rev.
∫ ⎜⎝
d Qrev.
= (ΔS) irrev. Path
T
Integration can be performed only on a reversible path.
4.
dQ
≤0
T
At the equilibrium state, the system is at the peak of the entropy hill. (Isolated)
5.
TdS = dU + pdV
6.
TdS = dH – Vdp
7.
Famous relation S = K ln W
3.
Clausius Inequality:
∫
Where K = Boltzmann constant
W = thermodynamic probability.
8.
General case of change of entropy of a Gas
p
V ⎫
⎧
S2 – S1 = m ⎨cv ln 2 + c p ln 2 ⎬
p1
V1 ⎭
⎩
Initial condition of gas p1 , V1, T1, S1 and
Final condition of gas p2 , V2, T2, S2
Page 70 of 265
Entropy
Chapter 7
Questions with Solution P. K. Nag
Q7.1.
On the basis of the first law fill in the blank spaces in the following table
of imaginary heat engine cycles. On the basis of the second law classify
each cycle as reversible, irreversible, or impossible.
Cycle
(a)
(b)
(c)
(d)
Temperature
Source
327°C
1000°C
750 K
700 K
Rate of Heat Flow
Sink
Supply
27°C
420 kJ/s
100°C …kJ/min
300 K
…kJ/s
300 K
2500
kcal/h
Rejection
230 kJ/s
4.2 MJ/min
…kJ/s
…kcal/h
Rate of
work
Output
…kW
… kW
26 kW
1 kW
Efficiency
65%
65%
—
(Ans. (a) Irreversible, (b) Irreversible, (c) Reversible, (d) Impossible)
Solution:
Cycle
Temperature
Rate of Heat Flow
Rate of
work
Efficiency
Remark
(a)
Source
327ºC
Sink
27ºC
Supply
420 kJ/s
Rejection
230 kJ/s
190kW
0.4523
ηmax = 50%,
irrev.possible
(b)
1000ºC
100ºC
12000
kJ/km
4.2 kJ/m
7800 kW
65%
ηmax=70.7%
irrev.possible
(c)
750 K
300 K
43.33 kJ/s
17.33 kJ/s
26 kW
60%
300 K
2500
kcal/h
1640
kcal/h
1 kW
4.4%
(d)
700 K
Q7.2
ηmax= 60% rev.
possible
ηmax=57%
irrev.possible
The latent heat of fusion of water at 0°C is 335 kJ/kg. How much does the
entropy of 1 kg of ice change as it melts into water in each of the
following ways:
(a) Heat is supplied reversibly to a mixture of ice and water at 0°C.
(b) A mixture of ice and water at 0°C is stirred by a paddle wheel.
(Ans. 1.2271 kJ/K)
1 × 335
Ice + Water
kJ/ K
Solution : (a) (ΔS) system = +
273
Q
= 1.227 kJ/K
273 K
(b) (ΔS) system
Page 71 of 265
Entropy
Chapter 7
273
=
∫ mc
P
273
dT
=0
T
W
Q7.3
Solution:
Two kg of water at 80°C are mixed adiabatically with 3 kg of water at
30°C in a constant pressure process of 1 atmosphere. Find the increase in
the entropy of the total mass of water due to the mixing process (cp of
water = 4.187 kJ/kg K).
(Ans. 0.0576 kJ/K)
If final temperature of mixing is Tf then
2 × c P (353 – Tf )
2 kg
3 kg
= 3 × c P ( Tf – 303)
80°C = 353 K
30°C = 303 K
I
II
or Tf = 323 K
(ΔS) system = (ΔS) I + (ΔS) II
323
=
∫
353
323
m1 cP
dT
dT
+ ∫ m1 cP
T 303
T
323
⎛ 323 ⎞
= 2 × 4.187 ln ⎜
⎟ + 3 × 4.187 × ln
303
⎝ 353 ⎠
= 0.05915 kJ/K
Q7.4
In a Carnot cycle, heat is supplied at 350°C and rejected at 27°C. The
working fluid is water which, while receiving heat, evaporates from
liquid at 350°C to steam at 350°C. The associated entropy change is 1.44
kJ/kg K.
(a) If the cycle operates on a stationary mass of 1 kg of water, how much
is the work done per cycle, and how much is the heat supplied?
(b) If the cycle operates in steady flow with a power output of 20 kW,
what is the steam flow rate?
(Ans. (a) 465.12, 897.12 kJ/kg, (b) 0.043 kg/s)
Solution:
If heat required for evaporation is Q kJ/kg then
Q
= 1.44
(a)
(350 + 273)
or Q = 897.12 kJ/kg
(273 + 27)
It is a Carnot cycle so η = 1 −
(350 + 273)
∴ W = η.Q = 465.12 kJ
P
20
•
•
kg/s = 0.043 kg/s
P = mW or m =
(b)
=
W
465.12
Q7.5
A heat engine receives reversibly 420 kJ/cycle of heat from a source at
327°C, and rejects heat reversibly to a sink at 27°C. There are no other
heat transfers. For each of the three hypothetical amounts of heat
rejected, in (a), (b), and (c) below, compute the cyclic integral of d Q / T .
Page 72 of 265
Entropy
Chapter 7
from these results show which case is irreversible, which reversible, and
which impossible:
(a) 210 kJ/cycle rejected
(b) 105 kJ/cycle rejected
(c) 315 kJ/cycle rejected
(Ans. (a) Reversible, (b) Impossible, (c) Irreversible)
Solution:
(a)
(b)
(c)
Q7.6
dQ
+420
210
−
=
=0
T
(327 + 273) (27 + 273)
∴ Cycle is Reversible, Possible
∫
dQ
420 105
= +
−
= 0.35
T
600 300
∴ Cycle is Impossible
∫
dQ
420 315
−
= +
= – 0.35
T
600 300
∴ Cycle is irreversible but possible.
∫
In Figure, abed represents a Carnot cycle bounded by two reversible
adiabatic and two reversible isotherms at temperatures T1 and T2 (T1 >
T2).
The oval figure is a reversible cycle, where heat is absorbed at
temperature less than, or equal to, T1, and rejected at temperatures
greater than, or equal to, T2. Prove that the efficiency of the oval cycle is
less than that of the Carnot cycle.
Page 73 of 265
Entropy
Chapter 7
Solution:
a
b
T1
P
d
c T2
V
Q7.7
Solution:
Water is heated at a constant pressure of 0.7 MPa. The boiling point is
164.97°C. The initial temperature of water is 0°C. The latent heat of
evaporation is 2066.3 kJ/kg. Find the increase of entropy of water, if the
final state is steam
(Ans. 6.6967 kJ/kg K)
(ΔS)Water
437.97
dT
= ∫ 1 × 4187 ×
T
273
p = 700 kPa
⎛ 437.97 ⎞
= 4.187 ln ⎜
⎟ kJ/ K
⎝ 273 ⎠
T = 437.97 K
= 1.979 kJ/K
(ΔS)Eva pour
T
273
K
1 × 2066.3
kJ/ K
437.97
= 4.7179 kJ/K
=
S
(Δs) system
= 6.697 kJ/kg – K
Q7.8
One kg of air initially at 0.7 MPa, 20°C changes to 0.35 MPa, 60°C by the
three reversible non-flow processes, as shown in Figure. Process 1: a-2
consists of a constant pressure expansion followed by a constant volume
cooling, process 1: b-2 an isothermal expansion followed by a constant
pressure expansion, and process 1: c-2 an adiabatic
Page 74 of 265
Entropy
Chapter 7
Expansion followed by a constant volume heating. Determine the change
of internal energy, enthalpy, and entropy for each process, and find the
work transfer and heat transfer for each process. Take cp = 1.005 and c v
= 0.718 kJ/kg K and assume the specific heats to be constant. Also assume
for air pv = 0.287 T, where p is the pressure in kPa, v the specific volume
in m3/kg, and T the temperature in K.
∴
p1 = 0.7 MPa = 700 kPa
v1 = 0.12013 m3/kg
∴
Ta = 666 K
∴
p 2 = 350 kPa
v 2 = 0.27306 m3/kg
Solution:
T1 = 293 K
p a = 700 kPa
v a = 0.27306 m3/kg
T2 = 333 K
For process 1–a–2
Q1 – a = ua - u1 +
va
∫ p dV
v1
= u a – u1 + 700(0.27306 – 0.12013)
= u a – u1 + 107
Qa – 2 = u 2 – u a + 0
∴
u a – u1 = 267.86 kJ/kg
Page 75 of 265
Entropy
Chapter 7
u 2 – u a = –239 kJ/kg
Q1 – a =
Ta
∫c
P
dT
T1
= 1.005 × (666 – 293)
= 374.865 kJ/kg
Qa – 2 =
T2
∫c
v
dT
Ta
= 0.718 (333 – 666)
= –239 kJ/kg
(i) Δu = u 2 - u1 = 28.766 kJ/kg
(ii) Δh = h2 – h1 = u 2 - u1 + p 2 v 2 – p1 v1
= 28.766 + 350 × 0.27306 – 700 × 0.12013 = 40.246 kJ/kg
(iii) Q = Q2 + Q1 = 135.865 kJ/kg
(iv) W = W1 + W2 = 107 kJ/kg
(v) Δs = s2 – s1 =
⎛T
= Cv ln ⎜ 2
⎝ Ta
( s2
– sa ) +
( sa –
s1 )
⎛ Ta ⎞
⎞
⎟ + CP ln ⎜ T ⎟
⎝ 1⎠
⎠
= 0.3275 kJ/kg – K
Q7.9
Ten grammes of water at 20°C is converted into ice at –10°C at constant
atmospheric pressure. Assuming the specific heat of liquid water to
remain constant at 4.2 J/gK and that of ice to be half of this value, and
taking the latent heat of fusion of ice at 0°C to be 335 J/g, calculate the
total entropy change of the system.
(Ans. 16.02 J/K)
Solution:
273
S2 – S1 =
m cP dT
T
293
∫
1
273
kJ/ K
293
= –0.00297 kJ/K
= –2.9694 J/K
− mL
S3 – S2 =
T
−0.01 × 335 × 1000
=
273
= –12.271 J/K
293 K
= 0.01 × 4.2 × ln
3
T
273 K
4
268 K
S
Page 76 of 265
2
Entropy
Chapter 7
268
m cP dT
268
⎛ 4.2 ⎞
= 0.01 × ⎜
kJ/ K
⎟ × ln
T
273
⎝ 2 ⎠
273
= –0.3882 J/K
S4 – S3 =
∴
∴
Q7.10
∫
S4 – S1 = – 15.63 J/K
Net Entropy change = 15.63 J/K
Calculate the entropy change of the universe as a result of the following
processes:
(a) A copper block of 600 g mass and with Cp of 150 J/K at 100°C is placed
in a lake at 8°C.
(b) The same block, at 8°C, is dropped from a height of 100 m into the
lake.
(c) Two such blocks, at 100 and 0°C, are joined together.
(Ans. (a) 6.69 J/K, (b) 2.095 J/K, (c) 3.64 J/K)
Solution:
281
(a)
dT
T
373
281
= 150 ln
J/ K
373
= –42.48 J/K
As unit of CP is J/K there for
∴
It is heat capacity
i.e.
Cp = m c p
(ΔS) copper =
∫ mc
P
(ΔS) lake =
C p (100 − 8)
J/ K
281
150(100 − 8)
J/ K = 49.11 J/K
=
281
(ΔS) univ = (ΔS) COP + (ΔS) lake = 6.63 J/K
(b) Work when it touch water = 0.600 × 9.81 × 100 J = 588.6 J
As work dissipated from the copper
(ΔS) copper = 0
As the work is converted to heat and absorbed by water then
W=Q
588.6
=
J/ K = 2.09466 J/K
(ΔS) lake =
281
281
∴
(ΔS) univ = 0 + 2.09466 J/k = 2.09466 J/K
100 + 0
(c) Final temperature (Tf) =
= 50º C = 323 K
2
Page 77 of 265
100 m
Entropy
Chapter 7
Tf
dT
(ΔS)I = C p ∫
;
T
T1
(ΔS)II = C p
Tf
dT
T
T2
∫
⎛T ⎞
⎛T ⎞
∴ (ΔS) system = 150 ln ⎜ f ⎟ + 150 ln ⎜ f ⎟
⎝ T1 ⎠
⎝ T2 ⎠
323
323
J/ K = 3.638 J/K
+ ln
= 150 ln
373
273
{
Q7.11
}
A system maintained at constant volume is initially at temperature T1,
and a heat reservoir at the lower temperature T0 is available. Show that
the maximum work recoverable as the system is cooled to T0 is
⎡
T ⎤
W = Cv ⎢(T1 − T0 ) − T0 ln 1 ⎥
T0 ⎦
⎣
Solution:
For maximum work obtainable the process should be reversible
T0
dT
⎛T ⎞
(ΔS)body = ∫ Cv
= Cv ln ⎜ 0 ⎟
T
⎝ T1 ⎠
T1
Q−W
T0
(ΔS)cycle = 0
⎛T ⎞ Q−W
(ΔS)univ. = Cv ln ⎜ 0 ⎟ +
≥0
T0
⎝ T1 ⎠
T1
Q1
(ΔS)resoir =
∴
∴
or
or
or
or
∴
⎛T ⎞ Q−W
≥0
Cv ln ⎜ 0 ⎟ +
T0
⎝ T1 ⎠
⎛T ⎞
Cv T0 ln ⎜ 0 ⎟ + Q − W ≥ 0
⎝ T1 ⎠
⎛T ⎞
W ≤ Q + Cv T0 ln ⎜ 0 ⎟
⎝ T1 ⎠
∴
Cv = mcv
H.E.
W
(Q1 – W)
T0
Q = Cv(T1 – T0)
⎛T ⎞
W ≤ Cv (T1 – T0) + Cv T0 ln ⎜ 0 ⎟
⎝ T1 ⎠
⎧⎪
⎛ T ⎞ ⎫⎪
W ≤ Cv ⎨(T1 − T0 ) + T0 ln ⎜ 0 ⎟ ⎬
⎝ T1 ⎠ ⎭⎪
⎩⎪
⎧⎪
⎛ T ⎞ ⎫⎪
Maximum work Wmax = Cv ⎨(T1 − T0 ) + T0 ln ⎜ 0 ⎟ ⎬
⎪⎩
⎝ T1 ⎠ ⎪⎭
Q7.12
If the temperature of the atmosphere is 5°C on a winter day and if 1 kg of
water at 90°C is available, how much work can be obtained. Take cv, of
water as 4.186 kJ/kg K.
Solution:
TRY PLEASE
Q7.13
A body with the equation of state U = CT, where C is its heat capacity, is
heated from temperature T1 to T2 by a series of reservoirs ranging from
Page 78 of 265
Entropy
Chapter 7
T1 to T2. The body is then brought back to its initial state by contact with
a single reservoir at temperature T1. Calculate the changes of entropy of
the body and of the reservoirs. What is the total change in entropy of the
whole system?
If the initial heating were accomplished merely by bringing the body
into contact with a single reservoir at T2, what would the various
entropy changes be?
Solution:
TRY PLEASE
Q7.14
A body of finite mass is originally at temperature T1, which is higher
than that of a reservoir at temperature T2. Suppose an engine operates in
a cycle between the body and the reservoir until it lowers the
temperature of the body from T1 to T2, thus extracting heat Q from the
body. If the engine does work W, then it will reject heat Q–W to the
reservoir at T2. Applying the entropy principle, prove that the maximum
work obtainable from the engine is
W (max) = Q – T2 (S1 – S2)
Where S1 – S2 is the entropy decrease of the body.
Solution:
If the body is maintained at constant volume having constant volume
heat capacity Cv = 8.4 kJ/K which is independent of temperature, and if
T1 = 373 K and T2 = 303 K, determine the maximum work obtainable.
(Ans. 58.96 kJ)
Final temperature of the body will be T2
∴
S2 – S1 =
T2
∫ mc
v
T1
dT
⎛T ⎞
= m cv ln ⎜ 2 ⎟
T
⎝ T1 ⎠
[ cv = heat energy CV]
(ΔS) reservoir =
Q−W
T2
∴ (ΔS) H.E. = 0
or
Q−W
≥0
T2
T2 (S2 – S1) + Q – W ≥ 0
or
W ≤ Q + T2 (S2 – S1)
or
W ≤ [Q – T2 (S1 – S2)]
∴
Wmax = [Q – T2 (S1 – S2)]
∴
(ΔS) univ. = (S2 − S1 ) +
Wmax = Q – T2 (S1 – S2)
⎛T ⎞
= Q + T2Cv ln ⎜ 2 ⎟
⎝ T1 ⎠
⎛T ⎞
= Cv (T1 – T2) + T2 CV ln ⎜ 2 ⎟
⎝ T1 ⎠
⎡
⎛ 303 ⎞ ⎤
= 8.4 ⎢373 − 303 + 303 ln ⎜
⎟⎥
⎝ 373 ⎠ ⎦
⎣
Page 79 of 265
Entropy
Chapter 7
= 58.99 kJ
Each of three identical bodies satisfies the equation U = CT, where C is
the heat capacity of each of the bodies. Their initial temperatures are
200 K, 250 K, and 540 K. If C = 8.4 kJ/K, what is the maximum amount of
work that can be extracted in a process in which these bodies are
brought sto a final common temperature?
(Ans. 756 kJ)
Q7.15
Solution:
U = CT
Therefore heat capacity of the body is C = 8.4 kJ/K
Let find temperature will be (Tf)
∴
W = W1 + W2
Q = Q1 + Q2
T
(ΔS) 540K body = C ln f kJ/ K
540
⎛ T ⎞
(ΔS) 250 K = C ln ⎜ f ⎟
⎝ 250 ⎠
⎛ T ⎞
(ΔS) 200 K = C ln ⎜ f ⎟
⎝ 200 ⎠
(ΔS) surrounds = 0
(ΔS)H.E. = 0
∴
540 K
Q
H.E.
Q1 – W1
250 K
W
Q2 – W 1
200 K
⎛
⎞
Tf3
(ΔS)univ. = C ln ⎜
⎟≥0
⎝ 540 × 250 × 200 ⎠
For minimum Tf
Tf3 = 540 × 250 × 200
∴ Tf = 300 K
∴
Q7.16
∴ Q = 8.4(540 – 300) = 2016 kJ
Q1 – W1 = 8.4(300 – 250) = 420 kJ
Q2 – W2 = 8.4(300 – 200) = 840 kJ
∴ Q1 + Q2 – (W1 + W2) = 1260
or (W1 + W2) = 2016 – 1260 kJ = 756 kJ
Wmax = 756 kJ
In the temperature range between 0°C and 100°C a particular system
maintained at constant volume has a heat capacity.
Cv = A + 2BT
With
A = 0.014 J/K and B = 4.2 × 10-4 J/K2
A heat reservoir at 0°C and a reversible work source are available. What
is the maximum amount of work that can be transferred to the reversible
work source as the system is cooled from 100°C to the temperature of the
reservoir?
(Ans. 4.508 J)
Solution:
Page 80 of 265
Entropy
Chapter 7
Find temperature of body is 273 K
373 K
273
∴
Q =
∫
273
C v dT = AT + BT2 ]373
373
Q
2
2
= –A(100) + B( 273 – 373 ) J
H.F.
= –28.532 J (flow from the system)
273
(ΔS) body =
∫
373
273
Cv
W
Q–W
273 K
dT
T
⎛ A + 2 BT ⎞
⎟ dT
T
⎠
373
273
= A ln
+ 2 B (273 − 373) J/ K
373
= –0.08837 J/K
=
∫ ⎜⎝
Q−W
; (ΔS)H.E. = 0
273
Q−W
≥0
(ΔS)univ = −0.08837 +
273
–24.125 + Q – W ≥ 0
W ≤ Q – 24.125
W ≤ (28.532 – 24.125) J
W ≤ 4.407 J
Wmax = 4.407 J
(ΔS)res. =
∴
or
or
or
or
Q7.17
(ΔS)surrounds = 0
Each of the two bodies has a heat capacity at constant volume
Cv = A + 2BT
Where
Solution:
Q7.18
A = 8.4 J/K and B = 2.1 × 10-2 J/K2
If the bodies are initially at temperatures 200 K and 400 K and if a
reversible work source is available, what are the maximum and
minimum final common temperatures to which the two bodies can be
brought? What is the maximum amount of work that can be transferred
to the reversible work source?
(Ans. Tmin = 292 K)
TRY PLEASE
A reversible engine, as shown in Figure during a cycle of operations
draws 5 MJ from the 400 K reservoir and does 840 kJ of work. Find the
amount and direction of heat interaction with other reservoirs.
200 K
300 K
Q3
Q2
E
W = 840 kJ
Page 81 of 265
400 K
Q1 = 5 MJ
Entropy
Chapter 7
Solution:
(Ans. Q2 = + 4.98 MJ and Q3 = – 0.82 MJ)
Let Q2 and Q3 both incoming i.e. out from the system
∴
Q2 → +ve,
Q3 → +ve
Q3
Q2
5000
(ΔS) univ =
+
+
+ ( Δ S)H.E. + ( Δ S)surrounds = 0
200 300 400
200 K
300 K
Q3
400 K
Q2
Q1 = 5 MJ
E
Or
or
W = 840 kJ
Q3 Q2 5000
+
+
+ 0+ 0 = 0
2
3
4
6 Q3 + 4 Q2 + 3 × 5000 = 0
Q3 + Q2 + 5000 – 840 = 0
Heat balance
4 Q3 + 4 Q2 + 16640 = 0
or
∴
(i) – (iii) gives
Q7.19
∴
2 Q3 = +1640
Q3 = +820 kJ
∴
Q2 = –4980 kJ
… (i)
… (ii)
… (iii)
(Here –ve sign means heat flow opposite to our assumption)
For a fluid for which pv/T is a constant quantity equal to R, show that
the change in specific entropy between two states A and B is given by
sB − sA =
∫
TB
TA
⎛ Cp ⎞
pB
⎜
⎟ dT − R ln
pA
⎝ T ⎠
A fluid for which R is a constant and equal to 0.287 kJ/kg K, flows
steadily through an adiabatic machine, entering and leaving through
two adiabatic pipes. In one of these pipes the pressure and temperature
are 5 bar and 450 K and in the other pipe the pressure and temperature
are 1 bar and 300 K respectively. Determine which pressure and
temperature refer to the inlet pipe.
(Ans. A is the inlet pipe)
For the given temperature range, cp is given by
Cp = a ln T + b
Where T is the numerical value of the absolute temperature and a = 0.026
kJ/kg K, b = 0.86 kJ/kg K.
(Ans. sB – s A = 0.0509 kJ/kg K. A is the inlet pipe.)
Page 82 of 265
Entropy
Chapter 7
Solution:
A
p
B
C dT R
+ dV
dS = v
T
V
pV
=R
T
V
R
∴
=
T
p
V
dQ = dH – Vdp
or
or
or
TdS = dH – Vdp
C dT Vdp
ds = P
−
T
T
CPdT R
− dp
ds =
T
p
Intrigation both side with respect A to B
SB
TB
PB
dp
⎛ CP ⎞
dT
R
d
s
=
−
∫S
∫T ⎜⎝ T ⎟⎠
∫P p
A
A
A
or
sB – s A
⎡ TB ⎛ CP ⎞
⎛ pB ⎞ ⎤
= ⎢∫ ⎜
⎟ dT − R ln ⎜
⎟ ⎥ proved
⎝ pA ⎠ ⎦⎥
⎣⎢ TA ⎝ T ⎠
300
sB – s A =
⎛ a lnT + b ⎞
⎛1⎞
⎟ dT − 0.287 × ln ⎜ ⎟
T
⎠
⎝5⎠
450
∫ ⎜⎝
300
⎡ (ln T)2
⎤
⎛1 ⎞
= ⎢a
+ b ln T ⎥ − 0.287 × ln ⎜ ⎟
2
⎣
⎦ 450
⎝5⎠
a
300
⎛1 ⎞
{(ln 300)2 − (ln 450)2 } + b ln
− 0.287 ln ⎜ ⎟
2
450
⎝5⎠
or sB – s A = 0.05094 kJ/kg – K
A is the inlet of the pipe
sB – s A =
∴
Q7.20
Two vessels, A and B, each of volume 3 m3 may be connected by a tube of
negligible volume. Vessel a contains air at 0.7 MPa, 95 ° C, while vessel B
contains air at 0.35 MPa, 205°C. Find the change of entropy when A is
connected to B by working from the first principles and assuming the
Page 83 of 265
Entropy
Chapter 7
Solution:
mixing to be complete and adiabatic. For air take the relations as given
in Example 7.8.
(Ans. 0.959 kJ/K)
Let the find temperature be (Tf)
p V
Mass of ( m A ) = A A
RTA
700 × 3
kg
=
0.287 × 368
= 19.88335 kg
A
B
0.7 MPa
700 kPa
368 K
350 kPa
478 K
Mass of gas ( m B ) =
Cp = 1.005 kJ/kg-K
cv = 0.718 kJ/kg-K
R = 0.287 kJ/kg-K
pB VB
350 × 3
= 7.653842 kg
=
R TB
0.287 × 478
For adiabatic mixing of gas Internal Energy must be same
∴
u A = m A c v TA
= 19.88335 × 0.718 × 368 kJ = 5253.66 kJ
u B = m B c v TB
Umixture
Or
= 7.653842 × 0.718 × 478 kJ = 2626.83 kJ
= ( m A c v + m B c v ) Tf
Tf = 398.6 K
If final pressure (pf)
∴
∴
∴
Q7.21
pf
× Vf = mf RTf
27.5372 × 0.287 × 398.6
kPa = 525 kPa
pf =
6
⎡
T
⎛ p ⎞⎤
(ΔS)A = m A ⎢c P ln f − R ln ⎜ f ⎟ ⎥ = 3.3277
TA
⎝ pA ⎠ ⎦
⎣
⎡
T
⎛ p ⎞⎤
(ΔS)B = m B ⎢ c P ln f − R ln ⎜ f ⎟ ⎥ = –2.28795 kJ/K
TB
⎝ pB ⎠ ⎦
⎣
(ΔS)univ = (ΔS)A + (ΔS)B + 0 = 0.9498 kJ/K
(a) An aluminium block (cp = 400 J/kg K) with a mass of 5 kg is initially
at 40°C in room air at 20°C. It is cooled reversibly by transferring
heat to a completely reversible cyclic heat engine until the block
reaches 20°C. The 20°C room air serves as a constant temperature
sink for the engine. Compute (i) the change in entropy for the block,
Page 84 of 265
Entropy
Chapter 7
(ii) the change in entropy for the room air, (iii) the work done by the
engine.
(b) If the aluminium block is allowed to cool by natural convection to
room air, compute (i) the change in entropy for the block, (ii) the
change in entropy for the room air (iii) the net the change in entropy
for the universe.
(Ans. (a) – 134 J/K, + 134 J/K, 740 J;
(b) – 134 J/K, + 136.5 J/K, 2.5 J/K)
Solution:
293
(a)
(ΔS) A1 =
m cP dT
T
313
∫
293
J/ K = –132.06 J/K
313
Q−W
(ΔS) air =
293
And Q = m c P (313 – 293) = 40000 J
5 × 400 × ln
313 K
5 kg
Q
H.E.
As heat is reversibly flow then
(ΔS)Al + (ΔS) air = 0
W
or
–132.06 + 136.52 –
=0
293
or
W = 1.306 kJ
•
Q–W
293 K
(b) (ΔS)Δf = Same for reversible or irreversible = –132.06 J/K
4000
(ΔS) air =
= 136.52 J/K
293
(ΔS) air = +4.4587 J/K
Q7.22
Two bodies of equal heat capacities C and temperatures T1 and T2 form
an adiabatically closed system. What will the final temperature be if one
lets this system come to equilibrium (a) freely? (b) Reversibly? (c) What
is the maximum work which can be obtained from this system?
Solution:
(a)
Freely Tf =
(b)
T1 + T2
2
Reversible
Let find temperature be Tf
the (ΔS)hot =
Tf
∫C
T1
dT
T
= C ln
(ΔS)cold =
Tf
∫C
T2
Tf
T1
dT
⎛T ⎞
= C ln ⎜ f ⎟
T
⎝ T2 ⎠
∴ (ΔS)univ. = (ΔS)hot = (ΔS)cold = (ΔS)surroundings
T
T
= C ln f + C ln f = 0
T1
T2
Page 85 of 265
T1
C
H.E.
Q–W
T2
W
W
Entropy
Chapter 7
or
Tf =
T1 T2
Q = C(T1 − Tf )
Q − W = C(Tf − T2 )
∴
−
+
−
W = C(T1 − Tf − Tf + T2 )
= C {T1 + T2 – 2 Tf }
= C[T1 + T2 − 2 T1T2 ]
Q7.23
Solution:
A resistor of 30 ohms is maintained at a constant temperature of 27°C
while a current of 10 amperes is allowed to flow for 1 sec. Determine the
entropy change of the resistor and the universe.
(Ans. ( Δ S) resistor = 0, ( Δ S) univ = 10 J/K)
If the resistor initially at 27°C is now insulated and the same current is
passed for the same time, determine the entropy change of the resistor
and the universe. The specific heat of the resistor is 0.9 kJ/kg K and the
mass of the resistor is 10 g.
(Ans. ( Δ S) univ = 6.72 J/K)
As resistor is in steady state therefore no change in entropy. But the work = heat
is dissipated to the atmosphere.
i 2 Rt
So (ΔS)atm =
Tatm
102 × 30 × 1
= 10 kJ/kg
300
If the resistor is insulated then no heat flow to
surroundings
So
(ΔS) surroundings = 0
=
And, Temperature of resistance (Δt)
102 × 30 × 1
= 333.33º C
=
900 × 0.01
∴
Final temperature (Tf) = 633.33 K
W=Q
Initial temperature (To) = 300 K
633.33
dT
∴
(ΔS) = ∫ m c
T
300
⎛ 633.33 ⎞
= 0.01 × 0.9 × ln ⎜
⎟ = 6.725 J/K
⎝ 300 ⎠
(ΔS)univ = (ΔS)rev. = 6.725 J/K
Q7.24
An adiabatic vessel contains 2 kg of water at 25°C. By paddle-wheel work
transfer, the temperature of water is increased to 30°C. If the specific
heat of water is assumed constant at 4.187 kJ/kg K, find the entropy
change of the universe.
(Ans. 0.139 kJ/K)
Page 86 of 265
Entropy
Chapter 7
Solution:
(ΔS)surr. = 0
(ΔS)sys =
303
∫ mc
298
dT
T
303
= 0.13934 kJ/K
298
+(ΔS)surr = 0.13934 + 0 = 0.13934 kJ/K
= 2 × 4.187 × ln
(ΔS)univ = ( ΔS)sys
∴
Q7.25
2 kg
298 K
303 K
A copper rod is of length 1 m and diameter 0.01 m. One end of the rod is
at 100°C, and the other at 0°C. The rod is perfectly insulated along its
length and the thermal conductivity of copper is 380 W/mK. Calculate the
rate of heat transfer along the rod and the rate of entropy production
due to irreversibility of this heat transfer.
(Ans. 2.985 W, 0.00293 W/K)
Solution:
0.01 m
1m
K = 380 W/m – K
373 K
•
Q = kA
A = 7.854 × 10–5 m2
273 K
ΔT
Δx
= 380 × 7.854 × 10 −5 ×
100
W = 2.9845 W
1
•
At the 373 K end from surrounding Q amount heat is go to the system. So at this
end
•
•
( Δ S)charge
Q
= −
373
•
And at the 273 K and from system Q amount of heat is rejected to the
surroundings.
∴
∴
Q7.26
•
•
( Δ S)charge
Q
=
273
•
•
Q
Q
( Δ S)univ. =
−
= 0.00293 W/K
273 373
•
A body of constant heat capacity Cp and at a temperature Ti is put in
contact with a reservoir at a higher temperature Tf. The pressure
remains constant while the body comes to equilibrium with the
reservoir. Show that the entropy change of the universe is equal to
Page 87 of 265
Entropy
Chapter 7
⎡ Ti − Tf
Cp ⎢
− ln
⎣⎢ Tf
⎛
Ti − Tf
⎜⎜ 1 +
Tf
⎝
⎞⎤
⎟⎟ ⎥
⎠ ⎦⎥
Prove that entropy change is positive.
x2 x3 x 4
+
..... –
{where x < 1}
Given ln (1 + x) = x –
2
3 4
Solution:
Final temperature of the body will be Tf
∴
(ΔS) resoier =
Tf
dT
⎛T ⎞
= C p ln ⎜ f ⎟
T
⎝ Ti ⎠
Ti
C p (Tf − T1 )
(ΔS) body = C p
∫
Tf
∴ Total entropy charge
T ⎤
⎡ T − Ti
+ ln f ⎥
(ΔS) univ = C p ⎢ f
T
Ti ⎦
f
⎣
T⎤
⎡ T − Ti
− ln i ⎥
= Cp ⎢ f
Tf ⎦
⎣ Tf
⎡ T − Ti
T − Tf
⎛
= Cp ⎢ f
− ln ⎜1 + i
Tf
⎝
⎣ Tf
Let
∴
∴
Tf
CP
Ti
⎞⎤
⎟⎥
⎠⎦
Tf − Ti
=x
as Tf > Ti
Tf
Tf − Ti
<1
Tf
(ΔS) in = CP {x – ln (1 + x)}
=
⎡
⎤
x
x3 x4
C p ⎢x − x +
−
+
+ .......... ........α ⎥
2
3
4
⎣
⎦
=
⎡ x2 x3 x 4 x5
⎤
Cp ⎢ −
+
−
+ ............ α ⎥
3
4
5
⎣2
⎦
=
⎡ x 2 (3 − 2 x) x 4 (5 − 4 x)
⎤
Cp ⎢
+
+ ....... α ⎥
6
20
⎣
⎦
2
∴
Q7.27
(ΔS) univ is +ve
An insulated 0.75 kg copper calorimeter can containing 0.2 kg water is in
equilibrium at a temperature of 20°C. An experimenter now places 0.05
kg of ice at 0°C in the calorimeter and encloses the latter with a heat
insulating shield.
(a) When all the ice has melted and equilibrium has been reached, what
will be the temperature of water and the can? The specific heat of
copper is 0.418 kJ/kg K and the latent heat of fusion of ice is 333
kJ/kg.
Page 88 of 265
Entropy
Chapter 7
(b)
Compute the entropy increase of the universe resulting from the
process.
(c) What will be the minimum work needed by a stirrer to bring back
the temperature of water to 20°C?
(Ans. (a) 4.68°C, (b) 0.00276 kJ/K, (c) 20.84 kJ)
Solution:
Mass of ice = 0.05 kg
(a) Let final temperature be ( Tf )
∴
0.75 × 0.418 × (293 – Tf )
+ 0.2 × 4.187 × (293 – Tf )
= 333 × 0.05 + 0.05 × 4.187
× ( Tf – 273)
or 1.1509(293 – Tf )
= 16.65 – 57.15255 + 0.20935 Tf
or 337.2137 – 1.1509 Tf
Wab = 0.2 kg
cv = 0.75 kJ /kg-K
T1 = 293 K
or Tf = 277.68 K = 4.68º C
(b) (ΔS)system
⎛ T ⎞
⎛ T ⎞
= 0.75 × 0.418 × ln ⎜ f ⎟ + 0.2 × 4.187 × ln ⎜ f ⎟
⎝ 293 ⎠
⎝ 293 ⎠
333 × 0.05
⎛ T ⎞
+
+ 0.05 × 4.187 ln ⎜ f ⎟
273
⎝ 273 ⎠
= 0.00275 kJ/K = 2.75 J/K
(c) Work fully converted to heat so no
Rejection.
∴
W = C × (20 – 4.68) = 20.84 kJ
∴
Q7.28
C = (Heat capacity) = 1.36025
Show that if two bodies of thermal capacities C1 and C2 at temperatures
T1 and T2 are brought to the same temperature T by means of a
reversible heat engine, then
ln T =
C1lnT1 + C2lnT2
C1 + C2
Solution:
T
(ΔS) 1 =
∫C
1
T1
dT
⎛T⎞
= C1 ln ⎜ ⎟
T
⎝ T1 ⎠
T
(ΔS) 2 =
∫C
T2
2
dT
⎛T⎞
= C2 ln ⎜ ⎟
T
⎝ T2 ⎠
(ΔS)univ = (ΔS)1 + (ΔS)2
For reversible process for an isolated system (ΔS) since.
⎛T⎞
⎛T⎞
0 = C1 ln ⎜ ⎟ + C2 ln ⎜ ⎟
⎝ T1 ⎠
⎝ T2 ⎠
Page 89 of 265
Entropy
Chapter 7
C
or
or
or
Solution:
T1
ln T =
C1
Q
H.E.
(C1 + C2) ln T = C1 ln T1 + C2 ln T2
or
Q7.29
C
1
2
⎛T⎞ ⎛T⎞
⎜T ⎟ ⎜T ⎟ = 1
⎝ 1⎠ ⎝ 2⎠
TC1 + C2 = T1C1 T2C2
W
Q–W
C1ln T1 + C2 ln T2
Proved
C1 + C2
T2
C2
Two blocks of metal, each having a mass of 10 kg and a specific heat of
0.4 kJ/kg K, are at a temperature of 40°C. A reversible refrigerator
receives heat from one block and rejects heat to the other. Calculate the
work required to cause a temperature difference of 100°C between the
two blocks.
Mass = 10 kg
C = 0.4 kJ/kg – K
T = 40º C = 313 K
⎛ T ⎞
∴ (ΔS) hot = mc ln ⎜ f ⎟
⎝ 313 ⎠
⎛ T − 100 ⎞
(ΔS) cold = m c ln ⎜ f
⎟
⎝ 313 ⎠
Tf
For minimum work requirement process must be reversible
T1 = 313 K
so (ΔS)univ = 0
Tf (Tf − 100)
= 0 = ln 1
(313)2
∴
ln
or
Tf2 − 100 Tf − 3132 = 0
or
100 ± 1002 + 4 × 3132
Tf =
2
Q+W
W
R
Q
T1 = 313 K
Tf – 100
= 367 K or (–267)
∴
∴
Q7.30
Q + W = 10 × 10.4 × (367 – 313) = 215.87 kJ
Q = 10 × 0.4 × (313 – 267) = 184 kJ
Wmin = 31.87 kJ
A body of finite mass is originally at a temperature T1, which is higher
than that of a heat reservoir at a temperature T2. An engine operates in
infinitesimal cycles between the body and the reservoir until it lowers
the temperature of the body from T1 to T2. In this process there is a heat
flow Q out of the body. Prove that the maximum work obtainable from
the engine is Q + T2 (S1 – S2), where S1 – S2 is the decrease in entropy of
the body.
Page 90 of 265
Entropy
Chapter 7
Solution:
Try please.
Q7.31
A block of iron weighing 100 kg and having a temperature of 100°C is
immersed in 50 kg of water at a temperature of 20°C. What will be the
change of entropy of the combined system of iron and water? Specific
heats of iron and water are 0.45 and 4.18 kJ/kg K respectively.
(Ans. 1.1328 kJ/K)
Let final temperature is tf ºC
∴
100 × 0.45 × (100 – tf) = 50 × 4.18 × (tf – 20)
100 – tf = 4.644 tf – 20 × 4.699
or
5.644 tf = 192.88
or
tf = 34.1732º C
∴
tf = 307.1732 K
Solution:
Q7.32
ENTROPY = (ΔS) iron + (ΔS) water
⎛ 307.1732 ⎞
⎛ 307.1732 ⎞
= 100 × 0.45 ln ⎜
⎟ + 50 × 4.180 × ln ⎜
⎟
⎝ 373 ⎠
⎝ 293 ⎠
= 1.1355 kJ/K
36 g of water at 30°C are converted into steam at 250°C at constant
atmospheric pressure. The specific heat of water is assumed constant at
4.2 J/g K and the latent heat of vaporization at 100°C is 2260 J/g. For
water vapour, assume pV = mRT where R = 0.4619 kJ/kg K, and
Cp
= a + bT + cT2, where a = 3.634,
R
b = 1.195 × 10-3 K-1 and c = 0.135 × 10-6 K-2
Calculate the entropy change of the system.
Solution:
(Ans. 277.8 J/K)
m = 36 g = 0.036 kg
T1 = 30ºC = 303 K
T2 = 373 K
T3 = 523 K
(ΔS) Water
⎛ 373 ⎞
= m cP ln ⎜
⎟ kJ/ K
⎝ 303 ⎠
= 0.03143 kJ/K
mL
(ΔS) Vaporization =
T2
0.036 × 2260
=
373
= 0.21812 kJ/K
523
dT
(ΔS) Vapor = ∫ m c p
T
373
523
= mR
a
∫ (T
T3
T
T2
T1
S
+ b + CT) dT
373
Page 91 of 265
Entropy
Chapter 7
523
⎡
CT2 ⎤
= mR ⎢a ln T + bT +
⎥
2 ⎦ 373
⎣
523
C
⎡
⎤
= mR ⎢a ln
+ b × (523 − 373) + (5232 − 3732 ) ⎥
373
2
⎣
⎦
= 0.023556 kJ/kg
(ΔS) System = (ΔS) water + (ΔS) vaporization + (ΔS) vapor = 273.1 J/K
Q7.33
Solution:
Q7.34
Solution:
Q7.35
A 50 ohm resistor carrying a constant current of 1 A is kept at a constant
temperature of 27°C by a stream of cooling water. In a time interval of 1s
(a) What is the change in entropy of the resistor?
(b) What is the change in entropy of the universe?
(Ans. (a) 0, (b) 0.167 J/K)
Try please.
A lump of ice with a mass of 1.5 kg at an initial temperature of 260 K
melts at the pressure of 1 bar as a result of heat transfer from the
environment. After some time has elapsed the resulting water attains the
temperature of the environment, 293 K. Calculate the entropy
production associated with this process. The latent heat of fusion of ice
is 333.4 kJ/kg, the specific heat of ice and water are 2.07 and 4.2 kJ/kg K
respectively, and ice melts at 273.15 K.
(Ans. 0.1514 kJ/K)
Try please.
An ideal gas is compressed reversibly and adiabatically from state a to
state b. It is then heated reversibly at constant volume to state c. After
expanding reversibly and adiabatically to state d such that Tb = Td, the
gas is again reversibly heated at constant pressure to state e such that Te
= Tc. Heat is then rejected reversibly from the gas at constant volume till
it returns to state a. Express Ta in terms of Tb and Tc. If Tb = 555 K and Tc
= 835 K, estimate Ta. Take γ = 1.4.
⎛
⎞
Tbγ +1
T
, 313.29 K ⎟
Ans.
=
⎜
a
γ
Tc
⎝
⎠
Solution:
⎛T ⎞
(ΔS) bc = Cv ln ⎜ c ⎟
⎝ Tb ⎠
⎛T ⎞
(ΔS) de = Cp ln ⎜ e ⎟
⎝ Tb ⎠
⎛T ⎞
(ΔS) ea = Cv ln ⎜ a ⎟
⎝ Tc ⎠
(ΔS) Cycles = 0
Page 92 of 265
Entropy
Chapter 7
or
⎛ Tc ⎞
⎜ ⎟
⎝ Tb ⎠
Or
( Tc )
γ
γ+ 1
Solution:
Q7.37
Solution:
Q7.38
(555)
8351.4
d
V=C
a
S
γ = 1.4 + Gas
= 313.286 K
Liquid water of mass 10 kg and temperature 20°C is mixed with 2 kg of
ice at – 5°C till equilibrium is reached at 1 atm pressure. Find the
entropy change of the system. Given: cp of water = 4.18 kJ/kg K, cp of ice =
2.09 kJ/kg K and latent heat of fusion of ice = 334 kJ/kg.
(Ans.190 J/K)
Try please.
A thermally insulated 50-ohm resistor carries a current of 1 A for 1 s. The
initial temperature of the resistor is 10°C. Its mass is 5 g and its specific
heat is 0.85 J g K.
(a) What is the change in entropy of the resistor?
(b) What is the change in entropy of the universe?
(Ans. (a) 0.173 J/K (b) 0.173 J/K)
Try please.
The value of cp for a certain substance can be represented by cp = a + bT.
(a) Determine the heat absorbed and the increase in entropy of a mass m
of the substance when its temperature is increased at constant
pressure from T1 to T2.
(b) Find the increase in the molal specific entropy of copper, when the
temperature is increased at constant pressure from 500 to 1200 K.
Given for copper: when T = 500 K, cp = 25.2 × 103 and when T = 1200 K,
cp = 30.1 × 103 J/k mol K.
⎛
⎡
⎡
⎤⎤ ⎞
T
b
⎜ (a) m ⎢a(T2 − T1 ) + T22 − T12 , m ⎢a ln 2 + b(T2 − T2 ) ⎥ ⎥ ; ⎟
Ans. ⎜
2
T1
⎣
⎦ ⎦⎥ ⎟
⎣⎢
⎜
⎟
(b) 24.7 kJ/k mol K ⎠
⎝
dQ = Cp dT
(
Solution:
e
C
Ta
.Ta . = Tbγ + 1
1.4 + 1
V=C
b
T
Ta =
Ta =
Q7.36
Tb = Td
⎛ Ta ⎞
⎜ ⎟ =1
⎝ Tc ⎠
Tbγ + 1
Tcγ
Given Tb = 555 K, Tc = 835 K,
∴
c
Tc = Te
=
or
⎛T ⎞
⎛T ⎞
(Cp + Cv ) l n ⎜ c ⎟ + Cv ln ⎜ a ⎟ = 0
⎝ Tb ⎠
⎝ Tc ⎠
⎛T ⎞
⎛T ⎞
( γ + 1) ln ⎜ c ⎟ + ln ⎜ a ⎟ = ln 1
⎝ Tb ⎠
⎝ Tc ⎠
p
∴
T2
∴ Q = m ∫ cP dT
T1
T2
T2
⎡
bT2 ⎤
= m ∫ (a + bT) dT = m ⎢aT +
⎥
2 ⎦ T1
⎣
T1
Page 93 of 265
)
Entropy
Chapter 7
or
b
⎡
⎤
= m ⎢a(T2 − T1 ) + (T22 − T12 ) ⎥
⎣
⎦
2
TdS = Cp dT
dT
or
dS = m c p
T
S2
T2
2
(a + bT)
dT
m
dS
=
c
=
m
∫S
∫1 p T
∫T T dT
1
1
T
⎡
⎤
(S2 – S1) = [a ln T + bT]TT12 = m ⎢a ln 2 + b(T2 − T1 ) ⎥
T1
⎣
⎦
For a and b find
⇒
25.2 = a + b × 500
30.1 = a + b × 1200
∴ b × 700 = 4.9
∴ b = 0.007 kJ/kg K ∴ a = 21.7 kJ/kg – K
⎡
⎤
⎛ 1200 ⎞
∴
S2 – S1 = ⎢21.7 ln ⎜
⎟ + 0.007 (1200 − 500)⎥ kJ/ K = 23.9 kJ/K
⎝ 500 ⎠
⎣
⎦
Page 94 of 265
Availability & Irreversibility
Chapter 8
8.
Availability & Irreversibility
Some Important Notes
1.
Available Energy (A.E.)
T
T ⎞
T ⎞
⎛
⎛
Wmax = Q1 ⎜1 − 0 ⎟ = m cP ∫ ⎜1 − 0 ⎟ dT
T⎠
T1 ⎠
⎝
T0 ⎝
= (T1 – T0) ΔS
= u1 – u2 – T0 ( s1 − s2 )
(For closed system), it is not (φ1 – φ2) because change of volume is present there.
= h1 − h 2 – T0 ( s1 − s2 )
(For steady flow system), it is (A1 – A2) as in steady state no change in volume is
CONSTANT VOLUME (i.e. change in availability in steady flow)
2.
Decrease in Available Energy
= T0 [ΔS′ – ΔS]
4
S
Take ΔS′ & ΔS both +Ve Quantity
T
Q1
S
3.
S
Availability function:
V2
A = h – T0s +
+ gZ
2
Availability = maximum useful work
For steady flow
Availability = A1 – A0 = (h1 – h0) – T0 ( s1 – s0 ) +
φ = u – T0s + p 0 V
V12
+ gZ
2
(∴V0 = 0, Z0 = 0)
For closed system
Availability = φ1 – φ0 = u1 – u 0 – T0 ( s1 – s 0 ) + p 0 (V1 – V0 )
Available energy is maximum work obtainable not USEFULWORK.
4.
Unavailable Energy (U.E.)
= T0 (S1 – S2)
5.
Increase in unavailable Energy = Loss in availability
= T0 (ΔS) univ.
Page 95 of 265
Availability & Irreversibility
Chapter 8
6.
Irreversibility
I = Wmax – Wactual
= T0(ΔS) univ.
7.
Irreversibility rate = I rate of energy degradation
•
2
Sgen =
•
∫ m dS
1
•
= rate of energy loss (W lost )
•
= T0 × Sgen
8.
for all processes
Wactual ⇒ dQ = du + d Wact
h1 +
this for closed system
dWact
V12
V2
dQ
+ g Z1 +
= h 2 + 2 + g Z2 +
this for steady flow
2
dm
2
dm
9.
Helmholtz function, F = U – TS
10.
Gibb’s function, G = H – TS
•
11.
Entropy Generation number (NS) =
Sgen
•
m cP
12.
13.
Second law efficiency
Minimum exergy intake to perform the given task (X min )
= η 1 /ηCarnot
ηII =
Actual exergy intake to perform the given task (X)
Xmin = W, if work is involved
T ⎞
⎛
= Q ⎜1 − 0 ⎟ if Heat is involved.
⎝
T⎠
To Calculate dS
p
V ⎤
⎡
i) Use S2 – S1 = m ⎢cv ln 2 + cP l n 2 ⎥
p1
V1 ⎦
⎣
For closed system
TdS = dU + pdV
dT p
+ dV
or
dS = m c v
T
T
dT
dV
+ mR
= m cv
T
V
2
2
2
dT
dV
dS
=
m
c
+
mR
v∫
∫1
∫
T
V
1
1
For steady flow system
TdS = dH – Vdp
dT V
− dp
or
dS = m c p
T
T
2
2
2
dT
dp
m
c
mR
dS
=
p∫
∫1
∫
T
p
1
1
But Note that
pV = mRT
V
mR
=
T
p
Page 96 of 265
Availability & Irreversibility
Chapter 8
And
TdS = dU + pdV
TdS = dH – Vdp
Both valid for closed system only
14.
In Pipe Flow Entropy generation rate
1
2
• kg/s
m
p, T1
Due to lack of insulation it may be
T1 > T2 for hot fluid T1 < T2 for cold fluid
•
•
Sgen = Ssys
•
Q
−
T0
•
•
= m(S2 − S1 ) −
∴
15.
m c p (T2 - T1 )
•
T0
•
Rate of Irreversibility (I) = T0 Sgen
Flow with friction
•
Decrease in availability = m RT0 ×
Δp
p1
Page 97 of 265
p, T2
Availability & Irreversibility
Chapter 8
Questions with Solution P. K. Nag
What is the maximum useful work which can be obtained when 100 kJ
are abstracted from a heat reservoir at 675 K in an environment at 288
K? What is the loss of useful work if
(a) A temperature drop of 50°C is introduced between the heat source
and the heat engine, on the one hand, and the heat engine and the
heat sink, on the other
(b) The source temperature drops by 50°C and the sink temperature
rises by 50°C during the heat transfer process according to the linear
dQ
= ± constant?
law
dT
(Ans. (a) 11.2 kJ, (b) 5.25 kJ)
Q8.1
Solution:
Entropy change for this process
ΔS =
−100
kJ/ K
675
= 0.14815 kJ/K
Wmax = (T – T0) ΔS
= (675 – 288) ΔS = 57.333 kJ
(a) Now maximum work obtainable
338 ⎞
⎛
′
Wmax
= 100 ⎜1 −
⎟
625 ⎠
⎝
= 45.92 kJ
∴ Loss of available work = 57.333 – 45.92
= 11.413 kJ
dQ
= ± constant
(b) Given
dT
Let dQ = ± mc P dT
∴ When source temperature is (675 – T) and
since temperature (288 + T) at that time if dQ
heat is flow then maximum. Available work
from that dQ is dW .
288 + T ⎞
⎛
∴ dWmax . = dQ ⎜1 −
675 − T ⎟⎠
⎝
288 + T ⎞
⎛
m cP dT
= ⎜1 −
675 − T ⎟⎠
⎝
50
288 + T ⎞
⎛
Wmax = m cP ∫ ⎜1 −
dT
675 − T ⎟⎠
0 ⎝
⎧ −288 − T
−963 + 675 − T ⎫
=
⎨
⎬
675 − T
⎩ 675 − T
⎭
Page 98 of 265
∴
675 K
Q = 100 kJ
H.E.
T = 50 K
T1 = 625 K
W
T2 = 338 K
T = 50 K
288 K
Availability & Irreversibility
Chapter 8
50
963 ⎫
⎧
= m c p ∫ ⎨1 + 1 −
⎬ dT
− T⎭
675
0 ⎩
⎧
⎛ 675 − 50 ⎞ ⎫
m c p ⎨2(50 − 0) + 963 ln ⎜
⎟⎬
⎝ 675 − 0 ⎠ ⎭
⎩
= 25.887 mc P kJ
=
(675 – T)
Q1
W
H.E.
m c p × 50 = 100 kJ
= 51.773 kJ
∴
(288 + T)
mc p = 2 kJ/K
∴ Loss of availability = (57.333 – 51.773) kJ
= 5.5603 kJ
Q 8.2
Solution:
In a steam generator, water is evaporated at 260°C, while the combustion
gas (cp = 1.08 kJ/kg K) is cooled from 1300°C to 320°C. The surroundings
are at 30°C. Determine the loss in available energy due to the above heat
transfer per kg of water evaporated (Latent heat of vaporization of
water at 260°C = 1662.5 kJ/kg).
(Ans. 443.6 kJ)
Availability decrease of gas
Agas = h1 – h2 – T0 ( s1 – s2 )
⎛ T1 ⎞
⎟
⎝ T2 ⎠
= mc p ( T1 – T2 )
– T0 mc p ln ⎜
T ⎤
⎡
= m cP ⎢(T1 − T2 ) − T0 ln 1 ⎥
T2 ⎦
⎣
∴ T1 = 1573 K; T2 = 593 K; T0 = 303 K
= m × 739.16 kJ
Availability increase of water
= (T1 – T0) ΔS
Aw
mL
= (T1 − T0 ) ×
T1
{
= 1 × 1662.5 1 −
303
533
}
= 717.4 kJ
For mass flow rate of gas (m)
mg cPg (T2 − T1 ) = mw × L
∴
mg × 1.08 × (1300 – 320) = 1 × 1662.5
•
mg = 1.5708 kg/ of water of evaporator
Agas = 1161.1 kJ
Loss of availability
•
= A gas − A w
= (1161.1 – 717.4) kJ
= 443.7 kJ
Page 99 of 265
Availability & Irreversibility
Chapter 8
Q 8.3
Solution:
Exhaust gases leave an internal combustion engine at 800°C and 1 atm,
after having done 1050 kJ of work per kg of gas in the engine (cp of gas =
1.1 kJ/kg K). The temperature of the surroundings is 30°C.
(a) How much available energy per kg of gas is lost by throwing away
the exhaust gases?
(b) What is the ratio of the lost available energy to the engine work?
(Ans. (a) 425.58 kJ, (b) 0.405)
Loss of availability
1073
(a) =
T ⎞
⎛
m c p dT ⎜1 − 0 ⎟
T⎠
⎝
303
∫
⎧
⎛ 1073 ⎞ ⎫
= 1 × 1.1 ⎨(1073 − 303) − 303 ln ⎜
⎟⎬
⎝ 303 ⎠ ⎭
⎩
= 425.55 kJ
425.55
(b) r =
= 0.40528
1050
Q 8.4
Solution:
A hot spring produces water at a temperature of 56°C. The water flows
into a large lake, with a mean temperature of 14°C, at a rate of 0.1 m3 of
water per min. What is the rate of working of an ideal heat engine which
uses all the available energy?
(Ans. 19.5 kW)
Maximum work obtainable
329
•
287 ⎞
⎛
Wmax = ∫ m c p ⎜1 −
⎟ dT
T ⎠
⎝
287
•
{
= V ρ c p (329 − 287) − 287 ln
{
Solution:
}
}
0.1
329
kW
× 1000 × 4.187 (329 − 287) − 287 ln
60
287
= 19.559 kW
0.2 kg of air at 300°C is heated reversibly at constant pressure to 2066 K.
Find the available and unavailable energies of the heat added. Take T0 =
30°C and cp = 1.0047 kJ/kg K.
(Ans. 211.9 and 78.1 kJ)
Entropy increase
2066
dT
2066
ΔS = S2 – S1 = ∫ m c p
= 0.2 × 1.0047 × ln
= 0.2577 kJ/K
T
573
573
Availability increases
A increase = h2 – h1 – T0 ( s2 – s1 )
=
Q8.5
329
287
= mc p ( T2 – T1 ) – T0 × 0.2577
= 1250.24 – 78.084
= 1172.2 kJ
Heat input = m c p (T2 – T1) = 1250.24 kJ
Unavailable entropy = 78.086 kJ
Page 100 of 265
Availability & Irreversibility
Chapter 8
Q8.6
Solution:
Eighty kg of water at 100°C are mixed with 50 kg of water at 60°C, while
the temperature of the surroundings is 15°C. Determine the decrease in
available energy due to mixing.
(Ans. 236 kJ)
m1 = 80 kg
m2 = 50 kg
T1 = 100º = 373 K
T2 = 60º C = 333 K
T0 = 288 K
m T + m2 T2
= 357.62 K
Let final temperature ( Tf ) = 1 1
m1 + m2
Availability decrease of 80 kg
373
T ⎞
⎛
Adec = ∫ m c p dT ⎜1 − 0 ⎟
T⎠
⎝
357.62
⎡
⎛ 373 ⎞ ⎤
= m cP ⎢(373 − 357.62) − 288 ln ⎜
⎟⎥
⎝ 357.62 ⎠ ⎦
⎣
= 1088.4 kJ
Availability increase of 50 kg water
357.62
T ⎞
⎛
Ain = ∫ m c p ⎜1 − 0 ⎟ dT
T⎠
⎝
333
⎡
⎛ 357.62 ⎞ ⎤
= m c p ⎢(357.62 − 333) − 288 ln ⎜
⎟
⎝ 333 ⎠ ⎥⎦
⎣
= 853.6 kJ
Availability loss due to mixing
= (1088.4 – 853.6) kJ
= 234.8 kJ
∴
Q8.7
Solution:
A lead storage battery used in an automobile is able to deliver 5.2 MJ of
electrical energy. This energy is available for starting the car.
Let compressed air be considered for doing an equivalent amount of
work in starting the car. The compressed air is to be stored at 7 MPa,
25°C. What is the volume of the tank that would be required to let the
compressed air have an availability of 5.2 MJ? For air, pv = 0.287 T,
where T is in K, p in kPa, and v in m3/kg.
(Ans. 0.228 m3)
Electrical Energy is high Grade Energy so full energy is available
∴
A electric = 5.2 MJ = 5200 kJ
Availability of compressed air
= u1 – u0 – T0 ( s1 – s0 )
A air
= m cv (T1 – T0) – T0 ( s1 – s0 )
( s1
– s0 ) = cv ln
W=
T0 R ln
p1
v
+ cp ln 1
p0
v0
= c p ln
T1
p
− R ln 1
p0
T0
p1
p0
⎛ 7000 ⎞
= 298 × 0.287 × ln ⎜
⎟
⎝ 100 ⎠
= 363.36 kJ/kg
Here T1 = T0 = 25º C = 298 K
Let atm
Given p1 = 7 MPa = 7000 kPa
Page 101 of 265
pr = 1 bar = 100 kPa
Availability & Irreversibility
Chapter 8
5200
kg = 14.311 kg
363.36
Specific volume of air at 7 MPa, 25ºC then
RT
0.287 × 298 3
v=
=
m /kg = 0.012218 m3/kg
7000
p
∴ Required storage volume (V) = 0.17485 m3
∴
Q8.8
Solution:
Required mass of air =
Ice is to be made from water supplied at 15°C by the process shown in
Figure. The final temperature of the ice is – 10°C, and the final
temperature of the water that is used as cooling water in the condenser
is 30°C. Determine the minimum work required to produce 1000 kg of ice.
Take cp for water = 4.187 kJ/kg K, cp for ice = 2.093 kJ/kg K, and latent
heat of fusion of ice = 334 kJ/kg.
(Ans. 33.37 MJ)
Let us assume that heat rejection temperature is (T0)
(i) Then for 15ºC water to 0º C water if we need WR work minimum.
Q
T2
Then (COP) = 2 =
WR
T0 − T2
(T − T2 )
or WR = Q2 0
T2
⎛T
⎞
= Q2 ⎜ 0 − 1 ⎟
⎝ T2
⎠
When temperature of water is
T if change is dT
Then dQ 2 = – mc P dT
∴
∴
(heat rejection so –ve)
⎛T
⎞
dWR = − m cP dT ⎜ 0 − 1 ⎟
⎝T
⎠
273
⎛T
⎞
WRI = − m cP ∫ ⎜ 0 − 1 ⎟ dT
T
⎠
288 ⎝
288
⎡
⎤
− (288 − 273) ⎥
= m cP ⎢T0 ln
273
⎣
⎦
288
⎡
⎤
= 4187 ⎢T0 ln
− 15 ⎥ kJ
273
⎣
⎦
(ii) WR required for 0º C water to 0 º C ice
Page 102 of 265
Availability & Irreversibility
Chapter 8
⎛T
⎞
WRII = Q2 ⎜ 0 − 1 ⎟
⎝ T2
⎠
⎛T
⎞
= mL ⎜ 0 − 1 ⎟
⎝ T2
⎠
⎛ T
⎞
= 1000 × 335 ⎜ 0 − 1 ⎟
⎝ 273
⎠
⎛ T
⎞
= 335000 ⎜ 0 − 1 ⎟ kJ
⎝ 273
⎠
(iii) WR required for 0º C ice to –10 º C ice.
When temperature is T if dT temperature decreases
dQ 2 = – mc p ice dT
∴
T0
⎞
− 1⎟
⎝T
⎠
∴
dWR = − m c p ice dT ⎛⎜
∴
WRII = m c p ice
273
1
4.187
kJ/kg
c p,water =
2
2
4.187 ⎡
273
⎤
= 1000 ×
− 10 ⎥
T0 ln
⎢
2 ⎣
263
⎦
273
⎡
⎤
− 10 ⎥ kJ
= 2093.5 ⎢T0 ln
263
⎣
⎦
∴
Total work required
∴
273
⎡
⎤
⎢⎣T0 ln 263 − (273 − 263) ⎥⎦
=
c p,ice
∴
Solution:
263
⎞
− 1 ⎟ dT = m c p ice
⎠
Let
∴
Q8.9
⎛ T0
∫ ⎜⎝ T
WR = (i) + (ii) + (iii)
= [1529.2 T0 – 418740] kJ
WR and T0 has linear relationship
15 + 30
T0 =
º C = 22.5ºC = 295.5 K
2
WR = 33138.6 kJ = 33.139 MJ
A pressure vessel has a volume of 1 m3 and contains air at 1.4 MPa, 175°C.
The air is cooled to 25°C by heat transfer to the surroundings at 25°C.
Calculate the availability in the initial and final states and the
irreversibility of this process. Take p0 = 100 kPa.
(Ans. 135 kJ/kg, 114.6 kJ/kg, 222 kJ)
Given Ti = 175ºC = 448 K
Tf = 25ºC = 298 K
Vf = 1 m3
Vi = 1 m3
pi = 1.4 MPa = 1400 kPa
p f = 931.25 kPa
Calculated Data:
p 0 = 101.325 kPa,
T0 = 298 K
c p = 1.005 kJ/kg – K, cV = 0.718 kJ/kg – K; R = 0.287 kJ/kg – K
∴
Mass of air (m) =
pi Vi
1400 × 1
= 10.8885 kg
=
RTi
0.287 × 448
Page 103 of 265
Availability & Irreversibility
Chapter 8
∴
∴
Final volume (V0) =
mRT0
p0
=
10.8885 × 0.287 × 298
= 9.1907 m3
101.325
Initial availability
Ai = φ1 – φ0
= u1 – u0 – T0 ( s1 – s0 ) + p0 (V1 – V0)
V
p ⎫
⎧
= mc v (T1 - T0 ) - T0 ⎨mc p ln 1 + mc v ln 1 ⎬ + p0 (V1 - V0 )
V
p
0
0 ⎭
⎩
⎡
1
= m ⎢0.718(448 − 298) − 298 1.005 ln
9.1907
⎣
⎤
1400
+ 0.718 ln
+ 101.325 (1 − 9.1907) ⎥ kJ
101.325
⎦
{
}
= 1458.58 kJ = 133.96 kJ/kg
Final Availability
Af = φ f – φ0
V
p ⎫
⎧
= m cv (Tf − T0 ) − T0 ⎨m cP ln f + m cv ln f ⎬ + p0 (Vf − V0 )
V0
p0 ⎭
⎩
mRTf
⎡
⎤
⎢ pf = V = 931.25 kPa and Tf = T0 ⎥
f
⎣
⎦
V
p ⎫
⎧
= 0 − T0 m ⎨cP ln f + cv ln f ⎬ + p0 (Vf − V0 )
V0
p0 ⎭
⎩
= (2065.7 – 829.92) kJ
= 1235.8 kJ = 113.5 kJ/kg
Irreversibility = Loss of availability
= (1458.5 – 1235.8) kJ = 222.7 kJ
∴
Q8.10
Air flows through an adiabatic compressor at 2 kg/s. The inlet conditions
are 1 bar and 310 K and the exit conditions are 7 bar and 560 K. Compute
the net rate of availability transfer and the irreversibility. Take T0 = 298
K.
(Ans. 481.1 kW and 21.2 kW)
Solution:
Mass flow rate (m) = 2 kg/s
pi = 1 bar = 100 kPa
•
p f = 7 bar = 700 kPa
Ti = 310 K
Calculated data:
•
Tf = 560 K
•
•
m RTi
m RTf
Vi =
= 1.7794 m3/s V f =
= 0.4592 m3/s
pi
pf
Availability increase rate of air= B2 – B1
•
= h2 – h1 – T0 ( s2 – s1 )
v
p ⎫
•
⎧
= m cP (T2 − T1 ) − T0 ⎨m cP ln 2 + m cv ln 2 ⎬
p1 ⎭
v1
⎩
Page 104 of 265
T0 = 298 K
Availability & Irreversibility
Chapter 8
v
p ⎫⎤
• ⎡
⎧
= m ⎢cP (T2 − T1 ) − T0 ⎨cP ln 2 + cv ln 2 ⎬⎥
p1 ⎭⎦
v1
⎩
⎣
= 2[251.25 – 10.682] kW
= 481.14 kW
i
Actual work required= m(h2 – h1 )
W = 2 × 251.25 kW = 502.5 kW
∴
Q8.11
Solution:
Irreversibility = Wact. – Wmin.
= (502.5 – 481.14) kW
= 21.36 kW
An adiabatic turbine receives a gas (cp = 1.09 and cv = 0.838 kJ/kg K) at 7
bar and 1000°C and discharges at 1.5 bar and 665°C. Determine the
second law and isentropic efficiencies of the turbine. Take T0 = 298 K.
(Ans. 0.956, 0.879)
T1 = 1273 K
R = c P – c v = 0.252
p1 = 7 bar = 700 kPa
T1
T
( c − cv ) T1
RT1
v1 =
= p
p1
p1
0.252 × 1273 3
m /kg
=
700
= 0.45828 m3/kg
∴
T2 = 938 K
T2
T2′
T0 = 298 K
RT2
= 1.57584 m3/kg
p2
Wactual = h1 – h2 = mc P ( T1 – T2 )
= 1 × 1.09 × (1273 – 938) kW = 365.15 kW
p
V ⎤
⎡
S2 – S1 = m ⎢cv ln 2 + c p ln 2 ⎥
V1 ⎦
p1
⎣
{
S2
∴
∴
∴
}
150
1.57584
+ 1.09 × ln
kW/ K
700
0.43828
= 0.055326 kW/K
T
− S′2 = m c p ln 2′ = S2 – S1 = 0.055326
T2
T
1 × 1.09 ln 2′ = 0.055326
T2
T2
= 1.05207
T2′
T2
938
= 891.6 K
T2′ =
=
1.05207
(1.05207 )
Page 105 of 265
= 1 × 0.838 ln
2
2′
S
p 2 = 1.5 bar = 150 kPa
v2 =
1
Availability & Irreversibility
Chapter 8
Isentropic work = h1 − h′2 = m c p (T1 − T2′ )
= 3 × 1.09(1273 – 891.6) kW = 415.75 kW
365.15
= 87.83%
∴
Isentropic efficiency =
415.75
Change of availability
ΔA = A1 – A2
= h1 – h2 – T0(S1 – S2 )
= mc P ( T1 – T2 ) + T0 ( S2 – S1 )
= 1 × 1.09 (1273 – 938) + 298(0.055326) kW= 381.64 kW
Minimum exergy required to perform the task
Actual availability loss
365.15
= 95.7%
=
381.64
∴ ηII =
Q8.12
Air enters an adiabatic compressor at atmospheric conditions of 1 bar,
15°C and leaves at 5.5 bar. The mass flow rate is 0.01 kg/s and the
efficiency of the compressor is 75%. After leaving the compressor, the air
is cooled to 40°C in an after-cooler. Calculate
(a) The power required to drive the compressor
(b) The rate of irreversibility for the overall process (compressor and
cooler).
(Ans. (a) 2.42 kW, (b) 1 kW)
Solution:
2
p1 = 1 bar = 100 kPa
T1 = 288 K
2S
•
m = 0.01 kg/s
RT
v1 = 1 = 0.82656 m3/kg
p1
313 K
p 2 = 5.5 bar = 550 kPa
288 K
T
1
S
For minimum work required
to compressor is isentropic
γ( p2 V2 − p1 V1 )
Wisentropic =
γ −1
γ −1
⎡
⎤
γ
p
γ
⎛
⎞
2
⎢
=
RT1 ⎜ ⎟
− 1⎥
⎢⎝ p1 ⎠
⎥
γ −1
⎣
⎦
0.4
⎡
⎤
1.4
⎛ 550 ⎞ 1.4
⎢
⎥ kJ/kg
× 0.287 × 288 ⎜
−
1
=
⎟
⎥⎦
0.4
⎣⎢⎝ 100 ⎠
∴
40°C
Actual work required
Page 106 of 265
= 181.55 kW/kg
Availability & Irreversibility
Chapter 8
181.55
= 242 kJ/kg
0.75
∴ Power required driving the compressor
Wact =
(a)
•
= m Wact = 2.42 kW
Extra work addede in 2′ to 2 is (242 – 181.55) = 60.85 kJ/kg
∴
If C p (T2 − T2′ ) = 60.85
60.85
= 529.25 K
1.005
Availability loss due to cooling
T2 = T2′ +
or
∴
529.25
=
∫
313
288 ⎞
⎛
1 × 1.005 ⎜1 −
⎟ dT
T ⎠
⎝
⎧
⎛ 529.25 ⎞ ⎫
= 1.005 ⎨(529.21 − 313) − 288 ln ⎜
⎟ ⎬ kJ/kg
⎝ 313 ⎠ ⎭
⎩
= 65.302 kJ/kg
∴
Total available energy loss
= (60.85 + 65.302) kJ/kg = 126.15 kJ/kg
∴
Power loss due to irreversibility = 1.2615 kW
Q8.13
In a rotary compressor, air enters at 1.1 bar, 21 ° C where it is
compressed adiabatically to 6.6 bar, 250°C. Calculate the irreversibility
and the entropy production for unit mass flow rate. The atmosphere is at
1.03 bar, 20°C. Neglect the K.E. changes.
(Ans. 19 kJ/kg, 0.064 kJ/kg K)
Solution:
p1 = 1.1 bar = 110 kPa
p2
T1 = 294 K
p 2 = 6.6 bar = 660 kPa
T2 = 523 K
p 0 = 103 kPa
p1
T
2
2S
T0 = 293 K
p0
2
Δs = s2 – s1 =
d p⎞
⎛ dh
−v
⎟
T
T ⎠
1
∫ ⎜⎝
T
p ⎤
⎡
= ⎢CP ln 2 − R ln 2 ⎥
T1
p1 ⎦
⎣
⎡
523
⎛ 660 ⎞ ⎤
− 0.287 ln ⎜
= ⎢1.005 ln
⎟⎥
294
⎝ 110 ⎠ ⎦
⎣
= 0.064647 kJ/kg – K = 64.647 J/kg – K
1
S
Minimum work required
Wmin = Availability increase
= h2 – h1 – T0 ( s2 – s1 )
= mc P ( T2 – T1 ) – T0 Δs
= 1 × 1.005 (523 – 294) – 293 × 0.064647= 211.2 kJ/kg
Page 107 of 265
Availability & Irreversibility
Chapter 8
Actual work required (Wact ) = 230.145 kJ/kg
∴ Irreversibility = T0 Δs
= 293 × 0.064647 = 18.942 kJ/kg
Q8.14
Solution:
In a steam boiler, the hot gases from a fire transfer heat to water which
vaporizes at a constant temperature of 242.6°C (3.5 MPa). The gases are
cooled from 1100 to 430°C and have an average specific heat, cp = 1.046
kJ/kg K over this temperature range. The latent heat of vaporization of
steam at 3.5 MPa is 1753.7 kJ/kg. If the steam generation rate is 12.6 kg/s
and there is negligible heat loss from the boiler, calculate:
(a) The rate of heat transfer
(b) The rate of loss of exergy of the gas
(c) The rate of gain of exergy of the steam
(d) The rate of entropy generation. Take T0 = 21°C.
(Ans. (a) 22096 kW, (b) 15605.4 kW
(c) 9501.0 kW, (d) 20.76 kW/K)
(a) Rate of heat transfer = 12.6 × 1752.7 kW = 22.097 MW
•
If mass flow rate at gas is mg
•
Then mg cPg (1100 – 430) = 22097
or
•
mg = 31.53 kg/s
1373
Loss of exergy of the gas =
Q8.15
Solution:
294 ⎞
⎛
•
mg cPg ⎜1 −
⎟ dT
T ⎠
⎝
703
∫
⎡
⎛ 1373 ⎞ ⎤
•
= mg cPg ⎢(1373 − 703) − 294 ln ⎜
⎟⎥
⎝ 703 ⎠ ⎦
⎣
= 15606 kJ/s = 15.606 MW
294 ⎞
⎛
•
Gain of exergy of steam = m w L w ⎜1 −
⎟ = 9.497 MW
515.4 ⎠
⎝
Irriversibility
Rate of entropy gas =
T0
= 20.779 kW/K
An economizer, a gas-to-water finned tube heat exchanger, receives 67.5
kg/s of gas, cp = 1.0046 kJ/kg K, and 51.1 kg/s of water, cp = 4.186 kJ/kg K.
The water rises in temperature from 402 to 469 K, where the gas falls in
temperature from 682 K to 470 K. There are no changes of kinetic energy
and p0 = 1.03 bar and T0 = 289 K. Determine:
(a) Rate of change of availability of the water
(b) The rate of change of availability of the gas
(c) The rate of entropy generation
(Ans. (a) 4802.2 kW, (b) 7079.8 kW, (c) 7.73 kW/K)
T ⎞
⎛
(a) Rate of charge of availability of water = Q ⎜1 − 0 ⎟
⎝
T⎠
469
•
289 ⎞
⎛
= ∫ m w c p w dT ⎜1 −
⎟
T ⎠
⎝
402
469 ⎤
⎡
= 51.1 × 4.186 × ⎢(469 − 402) − 289 ln
kW
402 ⎥⎦
⎣
Page 108 of 265
Availability & Irreversibility
Chapter 8
= 4.823 MW (gain)
(b) Rate of availability loss of gas
682
•
289 ⎞
⎛
= ∫ mg cPg ⎜1 −
⎟ dT
T ⎠
⎝
470
682 ⎤
⎡
= 67.5 × 1.0046 ⎢(682 − 470) − 289 ln
470 ⎥⎦
⎣
= 7.0798 MW
∴ (c)
•
Rate of irreversibility (I) = 2.27754 MW
•
I
= 7.8808 kW/K
∴
Entropy generation rate Sgas =
T0
The exhaust gases from a gas turbine are used to heat water in an
adiabatic counter flow heat exchanger. The gases are cooled from 260 to
120°C, while water enters at 65°C. The flow rates of the gas and water are
0.38 kg/s and 0.50 kg/s respectively. The constant pressure specific heats
for the gas and water are 1.09 and 4.186 kJ/kg K respectively. Calculate
the rate of exergy loss due to heat transfer. Take T0 = 35°C.
(Ans. 12.5 kW)
Tgi = 260º C = 533 K
Tw i = 65ºC = 338 K
Tw o = 365.7 K (Calculated)
Tgo = 120ºC = 393 K
•
Q8.16
Solution:
•
•
mg = 0.38 kg/s
cpg = 1.09 kJ/kg – K
m w = 0.5 kg/s
c Pw = 4.186 kJ/kg – K
To = 35º C = 308 K
To calculate Two from heat balance
•
•
mg cPg (Tgi − Tgo ) = m w cPw (Two − Twi )
∴
Two = 365.7 K
Loss rate of availability of gas
•
⎡
⎛ 533 ⎞ ⎤
= mg c p g ⎢(533 − 393) − 308 ln ⎜
⎟ = 19.115 kW
⎝ 393 ⎠ ⎥⎦
⎣
Rate of gain of availability of water
•
⎡
⎛ 365.7 ⎞ ⎤
= m w c p w ⎢(365.7 − 338) − 308 ln ⎜
⎟ ⎥ = 7.199 kW
⎝ 338 ⎠ ⎦
⎣
∴
Rate of exergy loss = 11.916 kW
Q8.17
The exhaust from a gas turbine at 1.12 bar, 800 K flows steadily into a
heat exchanger which cools the gas to 700 K without significant pressure
drop. The heat transfer from the gas heats an air flow at constant
pressure, which enters the heat exchanger at 470 K. The mass flow rate
of air is twice that of the gas and the surroundings are at 1.03 bar, 20°C.
Determine:
(a) The decrease in availability of the exhaust gases.
(b) The total entropy production per kg of gas.
(c) What arrangement would be necessary to make the heat transfer
reversible and how much would this increase the power output of
Page 109 of 265
Availability & Irreversibility
Chapter 8
Solution:
the plant per kg of turbine gas? Take cp for exhaust gas as 1.08 and
for air as 1.05 kJ/kg K. Neglect heat transfer to the surroundings and
the changes in kinetic and potential energy.
(Ans. (a) 66 kJ/kg, (b) 0.0731 kJ/kg K, (c) 38.7 kJ/kg)
T ⎞
⎛
(a) Availability decrease of extra gases = Q ⎜1 − 0 ⎟
⎝
T⎠
800
⎡
293 ⎞
⎛
⎛ 800 ⎞ ⎤
= ∫ m c p ⎜1 −
⎟ dT = 1 × 1.08 ⎢(800 − 700) − 293 ln ⎜
⎟⎥
T
⎝
⎠
⎝ 700 ⎠ ⎦
⎣
700
= 65.745 kJ/kg
H.F.
Gas
T
H.F.
Water
S
(b)
Exit air temperature Texit
2 m c pa ( Te – 470 ) = m × c pg ( 800 – 700 )
or
∴
Te = 521.5 K
Availability increases
521.5 ⎤
⎡
= 2 × 1.05 × ⎢(521.5 − 470) − 293 ln
= 44.257 kJ/kg
470 ⎥⎦
⎣
•
∴ Sgas = 73.336 J/K of per kg gas flow
For reversible heat transfer
(ΔS) univ = 0
(ΔS) Gas = –(ΔS) water
800
m × 1.08 ln
700
⎛ 470 ⎞
= −2 m × 1.05 × ln ⎜
⎟
⎝ To ⎠
To
= 0.068673
470
∴
To = 503.4 K
∴
Q1 = m × 1.08(800 – 700) = 108 kJ/kg
Q2 = 2m × 1.05 (503.4 – 470) = 70.162 kJ/kg of gas
[i.e. extra output]
W = Q1 – Q2 = 37.84 kJ/kg of gas flow
or
Q8.18
ln
An air preheater is used to heat up the air used for combustion by
cooling the outgoing products of combustion from a furnace. The rate of
flow of the products is 10 kg/s, and the products are cooled from 300°C to
Page 110 of 265
Availability & Irreversibility
Chapter 8
Solution:
200°C, and for the products at this temperature cp = 1.09 kJ/kg K. The
rate of air flow is 9 kg/s, the initial air temperature is 40°C, and for the
air cp = 1.005 kJ/kg K.
(a) What is the initial and final availability of the products?
(b) What is the irreversibility for this process?
(c) If the heat transfer from the products were to take place reversibly
through heat engines, what would be the final temperature of the
air?
What power would be developed by the heat engines? Take To = 300 K.
(Ans. (a) 85.97, 39.68 kJ/kg, (b) 256.5 kW,
(c) 394.41 K, 353.65 kW)
To calculate final air temperature ( Tf )
•
•
mg c p g (573 − 473) = ma c p a (Tf − 313)
10 × 1.09 (573 – 473) = 9 × 1.005 (Tf – 313)
Tf = 433.5 K
Or
(a)
Initial availability of the product
573 ⎤
⎡
= c p g ⎢(573 − 300) − 300 ln
300 ⎥⎦
⎣
= 85.97 kJ/kg of product
Final availability
473 ⎤
⎡
= c p g ⎢(473 − 300) − 300 ln
= 39.68 kJ/kg of product
300 ⎥⎦
⎣
∴
Loss of availability = 46.287 kJ/kg of product
Availability gain by air
⎡
⎛ 433.5 ⎞ ⎤
= c p g ⎢(433.5 − 313) − 300 ln ⎜
⎟ = 22.907 kJ/kg of air
⎝ 313 ⎠ ⎥⎦
⎣
(b)
∴ Rate of irreversibility
•
I = (10 × 46.287 – 22.907 × 9) kW= 256.7 kW
(c)
For reversible heat transfer
(ΔS) Univ = 0
Gas
∴ (ΔS) gas + (ΔS) air = 0
or
(ΔS) gas = –(ΔS) air
T1
T
•
⎛T ⎞
or mg c p g ln ⎜ f ⎟
⎝ Ti ⎠
•
⎛T ⎞
= ma c p a ln ⎜ f ⎟
⎝ Ti ⎠
Air
T2
S
473
⎛
⎞
or 10 × 1.09 ln ⎜
⎟
(
)
10
1.09
ln
73
573
×
4
⎝
⎠
Page 111 of 265
Availability & Irreversibility
Chapter 8
Tf
313
Tf = 394.4 = 399.4 K
= −9 × 1.005 × ln
or
•
•
∴ Q1 = mg c p g (Ti − Tf ) = 1090 kJ
•
Q2 = ma c p a (394.4 − 313) = 736.263 kJ
∴
Q8.19
Solution:
•
•
W = Q1 − Q2 = 353.74 kW output of engine.
A mass of 2 kg of air in a vessel expands from 3 bar, 70°C to 1 bar, 40°C,
while receiving 1.2 kJ of heat from a reservoir at 120°C. The environment
is at 0.98 bar, 27°C. Calculate the maximum work and the work done on
the atmosphere.
(Ans. 177 kJ, 112.5 kJ)
Maximum work from gas
= u1 – u2 – T0 ( s1 – s2 )
= m cv (T1 − T2 )
T
p ⎤
⎡
− T0 ⎢m cP ln 1 − mR ln 1 ⎥
T2
p2 ⎦
⎣
⎡
⎡
343
⎛ 3 ⎞⎤ ⎤
= 2 ⎢0.718(343 − 313) − 300 ⎢1.005 ln
− 0.287 ln ⎜ ⎟ ⎥ ⎥
313
⎝ 1 ⎠⎦ ⎦
⎣
⎣
= 177.07 kJ
Work done on the atmosphere = p0 (V2 – V1)
T
T ⎤
⎡
= 98 ⎢mR 0 − mR 1 ⎥
p2
p1 ⎦
⎣
T ⎤
⎡T
= 98 mR ⎢ 2 − 1 ⎥
⎣ p2 p1 ⎦
= 111.75 kJ
Q8.20
1
Q = 1.2 kJ/kg
T
2
2S
S
Air enters the compressor of a gas turbine at 1 bar, 30°C and leaves the
compressor at 4 bar. The compressor has an efficiency of 82%. Calculate
per kg of air
(a) The work of compression
(b) The reversible work of compression
(c) The irreversibility. For air, use
T2 s
⎛p ⎞
=⎜ 2⎟
T1 ⎝ p1 ⎠
γ −1/ γ
Where T2s is the temperature of air after isentropic compression and γ =
1.4. The compressor efficiency is defined as (T2s – T1) / (T2 – T1), where T2
is the actual temperature of air after compression.
(Ans. (a) 180.5 kJ/kg, (b) 159.5 kJ/kg (c) 21 kJ/kg)
Solution:
Page 112 of 265
Availability & Irreversibility
Chapter 8
2
p1 = 1 bar = 100 kPa
T1 = 30º C = 303 K
2S
T
p2 = 4 bar = 400 kPa
T2 = ? ηcom = 6.82
1
S
(b) Minimum work required for compression is isentropic work
γ −1
⎧⎪
⎫⎪
γ
⎛ p2 ⎞ γ
∴ WR =
mRT ⎨⎜ ⎟
− 1⎬
γ −1
⎪⎩⎝ p1 ⎠
⎪⎭
0.4
⎧
⎫⎪
1.4 × 1 × 0.287 × 303 ⎪⎨⎛ 400 ⎞ 1.4
⎬ = 147.92 kJ/kg
=
−
1
⎜
⎟
(1.4 − 1)
⎩⎪⎝ 100 ⎠
⎭⎪
147.92
= 180.4 kJ/kg
(a)
Actual work =
0.82
∴ Extra work 32.47 kJ will heat the gas from T2′ to T2
γ −1
Q8.21
Solution:
∴
⎛ p′ ⎞ γ
T2′
= ⎜ 2⎟
T1
⎝ p1 ⎠
32.47 = mc P (T2 − T2′ )
∴
(c)
T2 = 482.6 K
Irreversibility (I) = (180.4 – 147.92) kJ/kg = 32.48 kJ/kg
∴ T2′ = 450.3 K
A mass of 6.98 kg of air is in a vessel at 200 kPa, 27°C. Heat is transferred
to the air from a reservoir at 727°C. Until the temperature of air rises to
327°C. The environment is at 100 kPa, 17°C. Determine
(a) The initial and final availability of air
(b) The maximum useful work associated with the process.
(Ans. (a) 103.5, 621.9 kJ (b) 582 kJ)
mRT2
p1 = 200kPa
p2 =
= 400 kPa
V2
po = 100 kPa
T1 = 300 K
mRT1
V1 =
= 3.005 m3
P1
Vo = 5.8095 m3
(a)
T2 = 600 K
To = 290 K
V2 = V1 = 3.005 m3
m = 6.98 kg
Initial availability
Ai = u1 – u0 – T0 ( s1 – s0 ) +
p0 (V1 – V0)
T
p ⎤
⎡
= m cv (T1 − T0 ) − mT0 ⎢m c p ln 1 − R ln 1 ⎥ + p0 (V1 − V0 )
T0
p0 ⎦
⎣
= 6.98 × 0.718 (300 – 290) – 6.98 × 290
Page 113 of 265
Availability & Irreversibility
Chapter 8
⎡
300
⎛ 200 ⎞ ⎤
−6.98 × 290 ⎢1.005 ln
− 0.287 ln ⎜
⎟ ⎥ + 100(3.005– 5.8095)
290
⎝ 100 ⎠ ⎦
⎣
= 103.4 kJ
Final availability
T2
p ⎤
− R ln 2 ⎥ + p0 (V2 − V0 )
T0
p0 ⎦
⎣
= 6.98 × 0.718(600 – 290) – 6.98 × 290
⎡
600
⎛ 400 ⎞ ⎤
− 0.287 ln ⎜
× ⎢1.005 ln
⎟ ⎥ + 100 (3.005 − 5.8095)
290
⎝ 100 ⎠ ⎦
⎣
= 599.5 kJ
Af = m
(b)
⎡
c v (T2 – T0) – mT0 ⎢cP ln
Maximum useful work
= u2 – u1 – T0 ( s2 – s1 ) + p 0 (V2 – V1)
T
p ⎤
⎡
= m cv (T2 − T1 ) − T0 m ⎢c p ln 2 − R ln 2 ⎥ + p0 (V2 − V1 )
T1
p1 ⎦
⎣
= 6.98 × 0.718(600 – 300) – 300
⎡
600
⎛ 400 ⎞ ⎤
× 6.98 ⎢1.005 ln
− 0.287 ln ⎜
⎟ ⎥ + p0 × 0
300
⎝ 200 ⎠ ⎦
⎣
∴
V2 = V1
= 461.35 kJ
Heat transfer to the vessel
mRT
V
= m cv (T2 – T1) = 6.98 × 0.718 × (600 – 300) kJ
Q=
∫m c
v
dT
p=
= 1503.402 kJ
T ⎞
⎛
Useful work loss of reservoir = Q ⎜1 − 0 ⎟
T⎠
⎝
290 ⎞
⎛
= 1503.402 ⎜1 −
⎟
1000
⎝
⎠
= 1067.47 kJ
∴
Q8.22
Solution:
Air enters a compressor in steady flow at 140 kPa, 17°C and 70 m/s and
leaves it at 350 kPa, 127°C and 110 m/s. The environment is at 100 kPa,
7°C. Calculate per kg of air
(a) The actual amount of work required
(b) The minimum work required
(c) The irreversibility of the process
(Ans. (a) 114.4 kJ, (b) 97.3 kJ, (c) 17.1 kJ)
Minimum work required
T2 = 127ºC = 400 K
T1 = 290 K
T0 = 280 K
2
2
V − V1
w = h 2 − h1 − T0 (s2 − s1 ) + 2
2000
T
V 2 − V12
p ⎤
⎡
= m cP (T2 − T1 ) − mT0 ⎢cP ln 2 − R ln 2 ⎥ + 2
T1
2000
p1 ⎦
⎣
Page 114 of 265
Availability & Irreversibility
Chapter 8
400
350 ⎤
⎡
− 0.287 ln
= 1 × 1.005(400 − 290) − 1 × 280 ⎢1.005 ln
290
140 ⎥⎦
⎣
+
= 110.55 – 16.86 + 3.6 = 97.29 kJ/kg
1.1102 − 702
kJ
2000
Actual work required
V22 − V12
h
h
= 2− 1+
= (110.55 + 3.6) kJ = 114.15 kJ
2000
∴
Irreversibility of the process
= T0 ( s2 – s1 ) = T0(ΔS) univ = 16.86 kJ/kg
Q8.23
Solution:
Air expands in a turbine adiabatically from 500 kPa, 400 K and 150 m/s to
100 kPa, 300 K and 70 m/s. The environment is at 100 kPa, 17°C. Calculate
per kg of air
(a) The maximum work output
(b) The actual work output
(c) The irreversibility
(Ans. (a) 159 kJ, (b) 109 kJ, (c) 50 kJ)
Maximum work output
V 2 − V22
w = h1 − h 2 − T0 (s1 − s2 ) + 1
2000
T1
p ⎫ V 2 − V22
⎧
= CP (T1 − T2 ) − T0 ⎨CP ln
− R ln 1 ⎬ + 1
T2
p2 ⎭
2000
⎩
{
= 1.005(400 − 300) − 290 1.005 ln
}
400
500 1502 − 702
− 0.287 ln
+
200
100
2000
= 159.41 kJ/kg
V12 − V22
= 100.5 + 8.8 = 109.3 kJ/kg
2000
The irreversibility (I) = T0(ΔS) univ = 50.109 kJ/kg
Calculate the specific exergy of air for a state at 2 bar, 393.15 K when the
surroundings are at 1 bar, 293.15 K. Take cp = 1 and R = 0.287 kJ/kg K.
(Ans. 72.31 kJ/kg)
Exergy = Available energy
= h1 – h2 – T0 ( s1 – s2 )
Actual output = h1 − h 2 +
Q8.24
Solution:
Q8.25
T
p ⎤
⎡
= C p (T1 − T0 ) − T0 ⎢C p ln 1 − R ln 1 ⎥
T0
p0 ⎦
⎣
⎡
393.15
⎛ 2 ⎞⎤
− 0.287 ln ⎜ ⎟ ⎥ kJ/kg
= 1 × (393.15 − 293.15) − 293.15 ⎢1 × ln
293.15
⎝ 1 ⎠⎦
⎣
= 72.28 kJ/kg
Calculate the specific exergy of CO2 (cp = 0.8659 and R = 0.1889 kJ/kg K)
for a state at 0.7 bar, 268.15 K and for the environment at 1.0 bar and
293.15 K.
(Ans. – 18.77 kJ/kg)
Solution:
Page 115 of 265
Availability & Irreversibility
Chapter 8
Emin
Exergy = Available energy
h1 – h0 – T0 ( s1 – s2 )
T
p ⎤
⎡
= C p (T1 − T0 ) − T0 ⎢C p ln 1 − R ln 1 ⎥
T0
p0 ⎦
⎣
= 0.8659 (268.15 − 293.15) − 293.15
{
× 0.8659 ln
p0 = 100 kPa
T0 = 293.15 K
p
}
268.15
70
− 0.1889 ln
kJ/ kg
293.15
100
p1 = 70 kPa, T1 = 68.15 K
= –18.772 kJ/kg
Q8.26
V
A pipe carries a stream of brine with a mass flow rate of 5 kg/s. Because
of poor thermal insulation the brine temperature increases from 250 K at
the pipe inlet to 253 K at the exit. Neglecting pressure losses, calculate
the irreversibility rate (or rate of energy degradation) associated with
the heat leakage. Take T0 = 293 K and cp = 2.85 kJ/kg K.
(Ans. 7.05 kW)
Solution:
1
2
T.dT
•
m = 5 kg/s
p, 250 K
p, 253 K
T0 = 293 K
cp = 2.85 kJ/kg – K
Entropy generation rate
•
•
Sgas = Ssys
•
Q
−
T0
•
•
= m (S2 − S1 ) −
m cP (253 − 250)
T0
•
3⎤
⎡ T
= m cP ⎢ ln 2 − ⎥ kW/ K
⎣ T1 T0 ⎦
= 0.0240777 kW/K
•
•
Where, Q = − m c p (253 − 250)
•
–ve because Q flux from surroundings.
T
S2 – S1 = c p ln 2
T1
•
∴
I = rate of energy degradation
= rate of exergy loss
•
To Sgen = 293 × 0.0240777 kW = 7.0548 kW
Q8.27
In an adiabatic throttling process, energy per unit mass of enthalpy
remains the same. However, there is a loss of exergy. An ideal gas
flowing at the rate m is throttled from pressure p1 to pressure p2 when
the environment is at temperature T0. What is the rate of exergy loss due
to throttling?
Page 116 of 265
Availability & Irreversibility
Chapter 8
i
i
p1 ⎞
⎛
⎜ Ans. I = m RT0 ln
⎟
p2 ⎠
⎝
Solution:
Adiabatic throttling process h1 = h2
•
∴
Rate of entropy generation (Sgen )
•
•
•
Sgen = ( ΔS)sys + ( ΔS)surr.
•
= ( ΔS)sys + 0
•
= m(S2 − S1 )
•
⎛p ⎞
= m R ln ⎜ 1 ⎟
⎝ p2 ⎠
(as no heat interaction with surroundings)
TdS = dh – Vdp
dh
dp
V
mR
or dS =
−V
=
p
T
T
T
dp
dS = 0 − mR
p
2
or
S2 – S1 = − ∫ mR
1
∴
• kg/s
m
p1, T1
p2, T2
p
p
dp
= − mR ln 2 = mR ln 1
p
p1
p2
•
Irreversibility rate (I)
•
= T0 × Sgen
⎛p ⎞
= T0 × mR ln ⎜ 1 ⎟
⎝ p2 ⎠
⎛p ⎞
= mR T0 ln ⎜ 1 ⎟
⎝ p2 ⎠
Q8.28.
Solution:
Air at 5 bar and 20°C flows into an evacuated tank until the pressure in
the tank is 5 bar. Assume that the process is adiabatic and the
temperature of the surroundings is 20°C.
(a) What is the final temperature of the air?
(b) What is the reversible work produced between the initial and final
states of the air?
(c) What is the net entropy change of the air entering the tank?
(d) Calculate the irreversibility of the process.
(Ans. (a) 410.2 K, (b) 98.9 kJ/kg,
(c) 0.3376 kJ/kg K, (d) 98.9 kJ/kg)
If m kg of air is entered to the tank then the enthalpy of entering fluid is equal to
internal energy of tank fluid.
(a)
h=v
∴ CpT1 = CvT2
⎛C ⎞
or T2 = ⎜ p ⎟ T1 = γ T1
⎝ Cv ⎠
Page 117 of 265
Availability & Irreversibility
Chapter 8
=1.4 × 293 K = 410.2 K
V1
(b) Reversible work
W = pdV work
= p (V2 – V1)
= p (0 – V1)
= pV1
Q8.29
v1 =
RT1
= 0.168182 m3/kg
p1
A Carnot cycle engine receives and rejects heat with a 20°C temperature
differential between itself and the thermal energy reservoirs. The
expansion and compression processes have a pressure ratio of 50. For 1
kg of air as the working substance, cycle temperature limits of 1000 K
and 300 K and T0 = 280 K, determine the second law efficiency.
(Ans. 0.965)
Solution:
Let Q1 amount of heat is in input. Then actual Carnot cycle produces work
360 ⎞
⎛
W = Q1 ⎜1 −
⎟ = 0.7 Q1
1000 ⎠
⎝
If there is no temperature differential between inlet and outlet then from Q1 heat
input Carnot cycle produce work.
280 ⎞
⎛
Wmax = Q1 ⎜1 −
⎟ = 0.72549 Q1
1020 ⎠
⎝
W
0.7
=
= 0.965
∴
Second law efficiency ( ηII ) =
0.72549
Wmax
Q8.30
Solution:
Energy is received by a solar collector at the rate of 300 kW from a
source temperature of 2400 K. If 60 kW of this energy is lost to the
surroundings at steady state and if the user temperature remains
constant at 600 K, what are the first law and the second law efficiencies?
Take T0 = 300 K.
(Ans. 0.80, 0.457)
First law efficiency
300 − 60
=
= 0.8
300
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Availability & Irreversibility
Chapter 8
300 ⎞
⎛
(300 − 60) ⎜1 −
⎟
600 ⎠ = 0.457
⎝
Second law efficiency =
300 ⎞
⎛
300 ⎜1 −
⎟
2400 ⎠
⎝
Q8.31
For flow of an ideal gas through an insulated pipeline, the pressure
drops from 100 bar to 95 bar. If the gas flows at the rate of 1.5 kg/s and
has cp = 1.005 and cv = 0.718 kJ/kg-K and if T0 = 300 K, find the rate of
entropy generation, and rate of loss of exergy.
(Ans. 0.0215 kW/K, 6.46 kW)
Solution:
1
2
• = 1.5 kg/s
m
p1 = 100 bar
T0 = 300 K
cp = 1.005 kJ/kg – K
cv = 0.718 kJ/kg – K
Rate of entropy generation
•
•
Sgen = ( Δ S)sys
p2 = 95 bar
•
Q
−
T0
•
As it is insulated pipe so Q = 0
•
= ( ΔS)sys
•
TdS = dh – Vdp
= m(S2 − S1 )
Here h1 = h2 so dh = 0
•
⎛p ⎞
= m R ln ⎜ 1 ⎟
⎝ p2 ⎠
∴
⎛ 100 ⎞
= 1.5 × 0.287 × ln ⎜
⎟ kW/K
⎝ 95 ⎠
TdS = – Vdp
V
dp
T
2
2
p
dp
mR
dS
=
−
∫1
∫1 p = mR ln p12
dS = −
= 0.022082 kW/K
•
Rate of loss of exergy = Irreversibility rate (I)
•
To Sgen = 300 × 0.22082 = 6.6245 kW
Q8.32
The cylinder of an internal combustion engine contains gases at 2500°C,
58 bar. Expansion takes place through a volume ratio of 9 according to
pv1.38 = const. The surroundings are at 20°C, 1.1 bar. Determine the loss of
availability, the work transfer and the heat transfer per unit mass. Treat
the gases as ideal having R = 0.26 kl/kg-K and cv = 0.82 kJ/kg-K.
(Ans. 1144 kJ/kg, 1074 kJ/kg, – 213 kJ/kg)
Solution:
Page 119 of 265
Availability & Irreversibility
Chapter 8
1
1
T
p
2
2
V
p1 = 58 bar = 5800 kPa
v1 = 0.1243 m3/kg (calculating)
T1 = 2500ºC = 2773 K
RT
∴ v1 = m 1 = 0.1243 m3/kg
p1
p0 = 1.1 bar = 110 kPa
S
p2 = 279.62 kPa (calculated)
v 2 = 9 v1 = 1.11876 m3/kg
T2 = 1203.2 K (calculated)
T0 = 20ºC = 293 K
∴ c P = c v + R = 1.08 kJ/kg
W = 0.82 kJ/kg – K
n
R = 0.26 kJ/kg – K
1.38
p
⎛v ⎞
⎛v ⎞
∴ 2 = ⎜ 1 ⎟ or p2 = p1 ⎜ 1 ⎟
p1 ⎝ v 2 ⎠
⎝ v2 ⎠
=
p1
91.38
n −1
T2
1
⎛v ⎞
= ⎜ 1⎟
= 0.38
9
T1
⎝ v2 ⎠
T1
∴ T2 = 0.38
= 1203.2 K
9
⇒ Loss of availability
∴ φ1 − φ2
= (u1 − u2 ) – T0 ( s1 – s2 ) + p0 ( v1 – v 2 )
T
p ⎤
⎡
= Cv (T1 − T2 ) − T0 ⎢C p ln 1 − R ln 1 ⎥ + p0 (v1 − v 2 )
T
p
2
2⎦
⎣
⎡
2773
⎛ 5800 ⎞ ⎤
= 0.82(2773 – 1203.2) − 293 ⎢1.08 ln
− 0.26 ln ⎜
⎟⎥
1203.2
⎝ 279.62 ⎠ ⎦
⎣
+ 110(0.1243 – 1.11876) kJ/kg
= 1211 kJ/kg
p v − p2 v 2
Work transfer (W) = 1 1
= 1074 kJ/kg
n −1
dQ = du + dW
Q1 – 2 = Cv (T2 – T1) + W1 – 2
∴
= –1287.2 + 1074 = –213.2 kJ/kg
Q8.33
In a counterflow heat exchanger, oil (cp = 2.1 kJ/kg-K) is cooled from 440
to 320 K, while water (cp = 4.2 kJ/kg K) is heated from 290 K to
temperature T. The respective mass flow rates of oil and water are 800
Page 120 of 265
Availability & Irreversibility
Chapter 8
and 3200 kg/h. Neglecting pressure drop, KE and PE effects and heat loss,
determine
(a) The temperature T
(b) The rate of exergy destruction
(c) The second law efficiency
Take T0 = I7°C and p0 = 1 atm.
(Ans. (a) 305 K, (b) 41.4 MJ/h, (c) 10.9%)
Solution:
440 K
c p = 2.1 kJ/kg – K
0
From energy balance
800 kg/K
•
(a) m P c P (440 − 320)
T
•
= m w c Pw (T − 290)
cp
W
= 4.2 kJ/kg – K
3200 kg/K
320 K
∴ T = 290 + 15 = 305 K
200 K
•
•
•
T0 = 17°C = 290 K
p0 = 1 m = 101.325 kPa
(b) Sgen = ( ΔS)0 + ( ΔS)
•
= mo c p o ln
•
Tfo
T
+ m wc p w ln fw
Tio
Tiw
S
320 3200
305 ⎤
⎡ 800
⎢⎣ 3600 × 2.1 × ln 440 + 3600 × 4.2 × ln 290 ⎥⎦
= 0.039663 kW/K = 39.6634 W/K
∴
•
Rate of energy destruction = To × Sgen = 290 × 0.039663 kW
= 11.5024 kW = 41.4 MJ/K
(c)
Availability decrease of oil
= A1 – A2 = h1 – h2 – T0 ( s1 – s2 )
T ⎤
•
⎡
= m0 c p0 ⎢(T1 − T2 ) − T0 ln 1 ⎥
T2 ⎦
⎣
800
440 ⎤
⎡
=
× 2.1 × ⎢(440 − 320) − 290 ln
3600
320 ⎥⎦
⎣
= 12.903 kW
Availability decrease of water
A1 – A2 = h1 – h2 – T0 ( s1 – s2 )
∴
T ⎤
•
⎡
= m w c p w ⎢(T1 − T2 ) − T0 ln 1 ⎥
T2 ⎦
⎣
3200
305 ⎤
⎡
=
× 4.2 × ⎢(305 − 290) − 290 ln
kW = 1.4 kW
3600
290 ⎥⎦
⎣
Gain of availability
1.4
Second law efficiency ( ηII ) =
=
= 10.85%
Loss for that
12.903
Page 121 of 265
Page 122 of 265
Properties of Pure Substances
Chapter 9
9.
Properties of Pure Substances
Some Important Notes
1.
2.
Triple point
On p-T diagram
It is a Point.
On p-V diagram
It is a Line
On T-s diagram
It is a Line
On U-V diagram
It is a Triangle
Triple point of water
T = 273.16 K
= 0.01ºC
3.
5 atm
And T = 216.55 K = – 56.45º C that so why
sublimation occurred.
Critical Point
For water pc = 221.2 bar ≈ 225.5 kgf/cm2
Tc = 374.15ºC ≈ 647.15 K
vc = 0.00317 m3/kg
At critical point
h fg = 0;
4.
Entropy (S) = 0
Internal Energy (u) = 0
Enthalpy (h) = u + pV
= Slightly positive
Triple point of CO2
p
3.
p = 0.00612 bar
= 4.587 mm of Hg
v fg = 0;
Sfg = 0
Mollier Diagram
⎡∵ TdS = dh − vdp⎤
⎢
⎥
⎢∴ ⎛⎜ ∂ h ⎞⎟ = T
⎥
⎢⎣ ⎝ ∂ S ⎠ p
⎥⎦
∴ The slope of an isobar on the h-s co-ordinates is equal to the absolute saturation
temperature at that pressure. And for that isobars on Mollier diagram diverges from one
another.
⎛∂h⎞
Basis of the h-S diagram is ⎜
⎟ =T
⎝ ∂ S ⎠P
Page 123 of 265
Properties of Pure Substances
Chapter 9
5.
Dryness friction
x=
6.
mv
mv + ml
v = (1 – x) v f + x v g
v = v f + x v fg
u = (1 – x) uf + x ug
u = uf + x ufg
h = (1 – x) h f + x hg
h = h f + x h fg
s = (1 – x) sf + x sg
s = sf + x sfg
7.
Super heated vapour: When the temperature of the vapour is greater than the
saturation temperature corresponding to the given pressure.
8.
Compressed liquid: When the temperature of the liquid is less than the saturation
temperature at the given pressure, the liquid is called compressed liquid.
9.
In combined calorimeter
x = x1 × x 2
x1 = from throttle calorimeter
x 2 = from separation calorimeter
Page 124 of 265
Properties of Pure Substances
Chapter 9
Questions with Solution P. K. Nag
Q9.1
Complete the following table of properties for 1 kg of water (liquid,
vapour or mixture)
Solution:
p bar
a
b
c
d
e
f
g
h
i
j
tºC
v m3/kg
x/%
Superheat
0ºC
0
0
0
0
140
87.024
0
249.6
50
201.70
0.0563
35
25.22
100
1.0135
100º
0.001044
0
20
212.42 0.089668
90
1
99.6
1.343
79.27
10
320
0.2676
100
5
238.8ºC
0.4646
100
4
143.6
0.4400
95.23
40
500
0.0864
100
20
212.4ºC
0.1145
100
15
400
0.203
100
Calculations: For (a) ………… For (b)
h = hf + x hf g
For (c) v = v f + x(v g − v f )
For (d) s = sf + x sf g
∴x=
h kJ/kg
2565.3
419.04
2608.3
2207.3
3093.8
2937.1
2635.9
3445.3
2932.5
3255.8
⇒ s= sf + x sf g
s − sf
= 0.7927 ∴ h = h f + x h f g
sfg
v = v f + x(v fg − v f )
For (e) tsat = 180ºC
v = 0.258 +
20
( 0.282 − 0.258 ) ,
50
20
(3157.8 − 3051.2) ) = 3093.8
50
20
s = 7.123 +
(7.310 − 7.123 ) = 7.1978
50
0.4646 − 0.425
t = 200 +
× 50 = 238.8º C
0.476 − 0.425
38.8
h = 2855.4 +
(2960.7 − 2855.4) = 2937.1
50
h = 3051.2 +
For (f)
s kJ/
kg – K
8.353
1.307
5.94772
6.104
7.1978
7.2235
6.6502
7.090
6.600
7.2690
Page 125 of 265
Properties of Pure Substances
Chapter 9
38.8
(7.271 − 7.059) = 7
50
0.4400 = 0.001084 + x(0.462 – 0.001084)
∴ x = 09523
h = 604.7 + x × 2133, s = 1.7764 + x × 5.1179 = 6.6502
12.4
v = 0.111 +
t = 262.4ºC
(0.121 − 0.111),
50
12.4
h = 2902.5 +
(3023.5 − 2902.5) = 2932.5
50
12.9
(6.766 − 6.545) = 6.600
s = 6.545 +
50
s = 7.059 +
(g)
(i)
Q9.2
(a) A rigid vessel of volume 0.86 m3 contains 1 kg of steam at a pressure
of 2 bar. Evaluate the specific volume, temperature, dryness fraction,
internal energy, enthalpy, and entropy of steam.
(b) The steam is heated to raise its temperature to 150°C. Show the
process on a sketch of the p–v diagram, and evaluate the pressure,
increase in enthalpy, increase in internal energy, increase in
entropy of steam, and the heat transfer. Evaluate also the pressure
at which the steam becomes dry saturated.
(Ans. (a) 0.86 m3/kg, 120.23°C, 0.97, 2468.54 k/kg, 2640.54 kJ/kg, 6.9592 kJ/kg K
(b) 2.3 bar, 126 kJ/kg, 106.6 kJ/kg, 0.2598 kJ/kg K, 106.6 kJ/K)
Solution:
0.86 m3
= 0.86 m3/kg
1 kg
→ at 2 bar pressure saturated steam sp. Volume = 0.885 m3/kg
So it is wet steam and temperature is saturation temperature
= 120.2º C
v − vf
→ v = v f + x(v g − v f )
∴x=
vg − v f
(a) → Specific volume = Volume/mass =
0.86 − 0.001061
= 0.97172
0.885 − 0.001061
→ Internal energy (u) = h – pv = 2644 – 200 × 0.86 = 2472 kJ/kg
→ Here h = h f + x h f g = 504.7 + 0.97172 × 2201.6 = 2644 kJ/kg
=
→ s = sf + x s f g = 1.5301 + 0.97172 × 5.5967 = 6.9685 kJ/kg – K
(b)
T2 = 150ºC = 423 K
v2 = 0.86 m3/kg
Page 126 of 265
Properties of Pure Substances
Chapter 9
S
p
V
0.885 − 0.86
+ (2 + 0 − 2) = 2.0641 bar
pS = 2 +
0.885 − 0.846
v S = 0.86 m3/kg
0.0691
(121.8 − 120.2) + 120.2 = 121.23º C = 394.23 K
0.1
Path 2 – 5 are is super heated zone so gas law (obey)
pS v1
pv
= 2 2
[∴ v2 = v1]
∴
TS
T2
T
423
∴
p2 = 2 × pS =
× 2.0641 = 2.215 bar
TS
394.23
From Molier diagram ps = 2.3 bar, h2 = 2770 kJ/kg, s2 = 7.095
∴
Δh = 127 kJ/kg, Δs = 0.1265 kJ/kg – K,
u2 = h 2 – p2 v 2 = 2580
∴
Δq = u2 − u1 = 107.5 kJ/kg
TS =
Q9.3
Ten kg of water at 45°C is heated at a constant pressure of 10 bar until it
becomes superheated vapour at 300°C. Find the change in volume,
enthalpy, internal energy and entropy.
(Ans. 2.569 m3, 28627.5 kJ, 26047.6 kJ, 64.842 kJ/K)
Solution:
2
30°C
p
1
2
T
1
m = 10 kg
V
At state (1)
p1 = 10 bar = 1000 kPa
T1 = 45ºC = 318 K
S
ΔS
At state (2)
p2 = p1 = 10 bar
T2 = 300ºC
For Steam Table
Page 127 of 265
Properties of Pure Substances
Chapter 9
v1 = 0.001010 m3/kg
h1 = 188.4 kJ/kg
u1 = h1 − p1 v1 = 187.39 kJ/kg
s1 = 0.693 kJ/kg – K
∴
Change in volume
Enthalpy change
Internal Energy change
Entropy change
Q9.4
v 2 = 0.258 m3/kg
u2 = 2793.2 kJ/kg
h2 = 3051.2 kJ/kg
s2 = 7.123 kJ/kg – K
= m ( v 2 – v1 ) = 2.57 m3
= m(h 2 − h1 ) = 28.628 MJ
= m(u2 − u1 ) = 26.0581 MJ
= m ( s2 – s1 ) = 64.3 kJ/K
Water at 40°C is continuously sprayed into a pipeline carrying 5 tonnes
of steam at 5 bar, 300°C per hour. At a section downstream where the
pressure is 3 bar, the quality is to be 95%. Find the rate of water spray in
kg/h.
(Ans. 912.67 kg/h)
Solution:
3
1
2
m1
p1 = 5 bar
= 500 kPa
T1 = 300°C
h1 = 3064.2 kJ/kg
m2 = (m1 + h3)
p2 = 3 bar
T2 = 133.5°C
h2 = 561.4 + 0.95 × 2163.2
= 2616.44 kJ/kg
T3 = 40º C
h 3 = 167.6 kJ/kg
∴
For adiabatic steady flow
•
•
•
•
•
m1h1 + m3 h3 = m2 (h 2 ) = (m1 + m3 ) h 2
∴
∴
•
•
m1 (h1 − h 2 ) = m3 (h 2 − h3 )
•
•
m3 = m1
(h1 − h 2 )
(h 2 − h3 )
⎧ 3064.2 − 2616.44 ⎫
= 5000 × ⎨
⎬ kg/hr
⎩ 2616.44 − 167.6 ⎭
= 914.23 kg/hr.
Q9.5
A rigid vessel contains 1 kg of a mixture of saturated water and
saturated steam at a pressure of 0.15 MPa. When the mixture is heated,
the state passes through the critical point. Determine
(a) The volume of the vessel
(b) The mass of liquid and of vapour in the vessel initially
Page 128 of 265
Properties of Pure Substances
Chapter 9
(c) The temperature of the mixture when the pressure has risen to 3
MPa
(d) The heat transfer required to produce the final state (c).
(Ans. (a) 0.003155 m3, (b) 0.9982 kg, 0.0018 kg,
(c) 233.9°C, (d) 581.46 kJ/kg)
Solution:
3
30 bar
2
p
p1 = 1.5 bar
1
V
It is a rigid vessel so if we
(a) Heat this then the process will be constant volume heating. So the volume of
the vessel is critical volume of water = 0.00317 m3
(b)
v = v f + x(v g − v fg )
∴x=
v − vf
0.00317 − 0.001053
=
1.159 − 0.001053
vg − v f
∴ Mass of vapour = 0.0018282 kg
∴ Mass of water = 0.998172 kg
(c)
As it passes through critical point then at 3 MPa i.e. 30 bar also it will be wet
steam 50 temperatures will be 233.8ºC.
(d)
Required heat (Q) = (u2 − u1 )
= (h 2 − h1 ) − (p2 v 2 − p1 v1 )
= (h2 f + x 2h fg2 ) − (h1f + x1h fg1 ) − p2 { v f + x 2 (v g − v f )
}2
+ p1 {v f + x1 (v g − v f )1 }
v 2 = v f2 + x 2 (v g2 − v f2 )
∴
x2 =
v 2 − v f2
v g 2 − v f2
=
0.00317 − 0.001216
= 0.029885
0.0666 − 0.001216
∴ Q = (1008.4 + 0.029885 × 1793.9)
– (467.1 + 0.0018282 × 2226.2) – 3000 (0.001216 +
0.029885 (0.0666 – 0.001216)) + 150(0.001053 + 0.001828 (1.159 – 0.0018282))
= 581.806 kJ/kg
Q9.6
A rigid closed tank of volume 3 m3 contains 5 kg of wet steam at a
pressure of 200 kPa. The tank is heated until the steam becomes dry
saturated. Determine the final pressure and the heat transfer to the
tank.
(Ans. 304 kPa, 3346 kJ)
Page 129 of 265
Properties of Pure Substances
Chapter 9
Solution:
V1 = 3 m3
m = 5 kg
∴
∴
3
= 0.6 m3/kg
5
p1 = 200 kPa = 2 bar
v − vf
(0.6 − 0.001061)
=
x1 = 1
= 0.67758
vg − v f
(0.885 − 0.001061)
v1 =
h1 = h f + x1 h fg = 504.7 + 0.67758 × 2201.6 = 1996.5 kJ/kg
u1 = h1 − p1 v1 = 1996.5 – 200 × 0.6 = 1876.5 kJ/kg
As rigid tank so heating will be cost vot heating.
∴
v g2 = 0.6 m3/kg
From Steam Table vg = 0.606 m3/kg
vg = 0.587 m3/kg
Q9.7
Solution:
for p = 300 kPa
for p = 310 kPa
10 × 0.006
= 303.16 kPa
0.019
∴
For V = 0.6 m3
∴
∴
u2 = h2 − p2 v 2 = 2725 – 303.16 × 0.6 = 2543 kJ/kg
Heat supplied Q = m(u2 − u1 ) = 3333 kJ
p2 = 300 ×
Steam flows through a small turbine at the rate of 5000 kg/h entering at
15 bar, 300°C and leaving at 0.1 bar with 4% moisture. The steam enters
at 80 m/s at a point 2 in above the discharge and leaves at 40 m/s.
Compute the shaft power assuming that the device is adiabatic but
considering kinetic and potential energy changes. How much error
would be made if these terms were neglected? Calculate the diameters of
the inlet and discharge tubes.
(Ans. 765.6 kW, 0.44%, 6.11 cm, 78.9 cm)
5000
•
kg/s
m = 5000 kg/hr =
3600
p1 = 15 bar
t1 = 300º C
1
2m
Z0
∴ From Steam Table
h1 = 3037.6 kJ/kg
V1 = 80 m/s
Z1 = (Z0 + 2) m
2
p2 = 0.1 bar
(100 − 4)
x2 =
= 0.96
100
t 2 = 45.8º C
Page 130 of 265
Properties of Pure Substances
Chapter 9
v1 = 0.169 m3/kg
h 2 = h f + x 2 h fg
= 191.8 + 0.96 × 2392.8 = 2489 kJ/kg
∴
V2 = 40 m/s, Z2 = Z0 m
v 2 = 14.083 m3/kg
V 2 − V22 g(Z1 − Z2 ) ⎤
• ⎡
Work output (W) = m ⎢(h1 − h 2 ) + 1
+
⎥
2000
2000 ⎦
⎣
5000 ⎡
802 − 902 9.81(2) ⎤
(3037.6
2489)
=
−
+
+
⎢
⎥ kW
3600 ⎣
2000
2000 ⎦
= 765.45 kW
If P.E. and K.E. is neglected the
∴
•
∴
W′ = m(h1 − h 2 ) = 762.1 kW
W − W′
Error =
× 100% = 0.44%
W
•
Area at inlet (A1) =
mv1
= 0.002934 m2 = 29.34 cm2
V1
∴ d1 = 6.112 cm
•
Area at outlet (A2) =
Q9.8
Solution :
mv 2
= 0.489 m2
V2
∴ d2 = 78.9 cm
A sample of steam from a boiler drum at 3 MPa is put through a
throttling calorimeter in which the pressure and temperature are found
to be 0.1 MPa, 120°C. Find the quality of the sample taken from the
boiler.
(Ans. 0.951)
p1 = 3 MPa = 30 bar
p2 = 0.1 MPa = 1 bar
t 2 = 120º C
20
(2776.4 − 2676.2)
h 2 = 2676.2 +
50
= 2716.3 kJ/kg
1
2
∴
h1 = h 2
h
∴
h1 = 2716.3 at 30 bar
If dryness fraction is
∴ h1 = hg1 + x h fg1
∴ x=
=
Q9.9
h1 − h f1
h fg1
S
2716.3 − 1008.4
= 0.952
1793.9
It is desired to measure the quality of wet steam at 0.5 MPa. The quality
of steam is expected to be not more than 0.9.
(a) Explain why a throttling calorimeter to atmospheric pressure will
not serve the purpose.
Page 131 of 265
Properties of Pure Substances
Chapter 9
(b)
Solution:
Will the use of a separating calorimeter, ahead of the throttling
calorimeter, serve the purpose, if at best 5 C degree of superheat is
desirable at the end of throttling? What is the minimum dryness
fraction required at the exit of the separating calorimeter to satisfy
this condition?
(Ans. 0.97)
(a)
After throttling if pressure is atm. Then minimum temperature required
t = tsat + 5ºC = 100 + 5 = 105º C
Then Enthalpy required
5
(2776.3 − 2676) kJ/kg = 2686 kJ/kg
= 2676 +
50
If at 0.5 MPa = 5 bar dryness fraction is < 0.9
∴
hmax = hf + 0.9 hfg = 640.1 + 0.9 × 2107.4 = 2536.76 kJ/kg
So it is not possible to give 5º super heat or at least saturation i.e. (2676 kJ/kg) so
it is not correct.
(b) Minimum dryness fraction required at the exit of the separating calorimeter
(x) then
2686 − 640.1
∴x=
= 0.971
h = h f + x h fg
2107.4
Q9.10
Solution:
The following observations were recorded in an experiment with a
combined separating and throttling calorimeter:
Pressure in the steam main–15 bar
Mass of water drained from the separator–0.55 kg
Mass of steam condensed after passing through the throttle valve –4.20
kg
Pressure and temperature after throttling–1 bar, 120°C
Evaluate the dryness fraction of the steam in the main, and state with
reasons, whether the throttling calorimeter alone could have been used
for this test.
(Ans. 0.85)
p1 = 15 bar = p2
T1 = 198.3º C = t 2
p3 = 1 bar, T3 = 120º C
∴
h3 = 2716.3 kJ/kg
1
2
1
2
1 bar
t = 120°C
3
3
4.2 kg
mw = 0.55 kg
h 2 = h 2f + x 2 × h fg2 = 844.7 + x 2 × 1945.2
∴
∴
x 2 = 0.96216
Total dryness fraction (x)
∴ dry steam = x 2 × 4.2
Page 132 of 265
Properties of Pure Substances
Chapter 9
x 2 × 4.2
0.96216 × 4.2
= 0.85
=
4.2 + 0.55
4.2 + 0.55
h1 = h f 1 + x h fg1 = 844.7 + 0.85 × 1945.2 = 2499.6 kJ/kg
=
But at 1 bar minimum 5º super heat i.e. 105ºC enthalpy is 2686 kJ/kg
So it is not possible to calculate only by throttling calorimeter.
Q9.11
Solution:
Steam from an engine exhaust at 1.25 bar flows steadily through an
electric calorimeter and comes out at 1 bar, 130°C. The calorimeter has
two 1 kW heaters and the flow is measured to be 3.4 kg in 5 min. Find the
quality in the engine exhaust. For the same mass flow and pressures,
what is the maximum moisture that can be determined if the outlet
temperature is at least 105°C?
(Ans. 0.944, 0.921)
30
(2776.4 − 2676.2) = 2736.3 kJ/kg
h 2 = 2676.2 +
50
•
•
•
m h1 = m h 2 − Q
1 1.25 bar
1 bar
2
130°C
• = 3.4 kg
m
5 mm
= 0.0113333 kg/s
2 kW capacity
•
h1 = h 2 −
Q
•
m
= 2560 kJ/kg
At 1.25 bar: from Steam Table
At 1.2 bar, hf = 439.4 kJ/kg
At 1.3 bar, hf = 449.2 kJ/kg
At 1.25 bar hf = 444.3 kJ/kg;
If dryness fraction is x
Then 2560 = 444.3 + x × 2241
or
x = 0.9441
If outlet temperature is 105º C then
h2 = 2686 kJ/kg
hfg = 2244.1 kJ/kg
hfg = 2237.8 kJ/kg
hfg = 2241 kJ/kg
(then from problem 9.9)
•
∴
h1 = h 2 −
Q
•
m
= 2509.53 kJ/kg
Then if dryness fraction is x2 then
∴ x 2 = 0.922 (min)
2509 = 444.3 + x 2 × 2241
Q9.12
Steam expands isentropically in a nozzle from 1 MPa, 250°C to 10 kPa.
The steam flow rate is 1 kg/s. Find the velocity of steam at the exit from
the nozzle, and the exit area of the nozzle. Neglect the velocity of steam
at the inlet to the nozzle.
Page 133 of 265
Properties of Pure Substances
Chapter 9
Solution:
The exhaust steam from the nozzle flows into a condenser and flows out
as saturated water. The cooling water enters the condenser at 25°C and
leaves at 35°C. Determine the mass flow rate of cooling water.
(Ans. 1224 m/s, 0.0101 m2, 47.81 kg/s)
At inlet
h1 = 2942.6 kJ/kg
t 2 = 45.8º C
s1 = 6.925 kJ/kg-K
s2 = 6.925 kJ/kg-K
If dry fraction x
h f 2 = 191.8 kJ/kg
v 2 = 12.274 m3/kg
1
2
p1 = 1000 kPa
= 10 bar
p2 = 10 kPa
= 0.1 bar
∴
∴
∴
t1 = 250°C m = 1 kg/s
V1 = 0
V2
s1 = s2 = 0.649 + x × 7.501
∴ x = 0.8367
h2 = 191.8 + 0.8367 × 2392.8 = 2193.8 kJ/kg
V2 = 2000(h1 − h 2 ) = 1224 m/s
∴
Outlet Area =
•
Q9.13
Solution:
∴
∴
mv 2
= 100.3 cm2
V2
If water flow rate is m kg/s
1 × (2193.8 – 191.8) = m 4.187 (35 – 25)
∴
m = 47.815 kg/s
⋅
A reversible polytropic process, begins with steam at p1 = 10 bar, t1 =
200°C, and ends with p2 = 1 bar. The exponent n has the value 1.15. Find
the final specific volume, the final temperature, and the heat transferred
per kg of fluid.
p1 = 10 bar = 1000 kPa
p2 = 1 bar = 100 kPa
t1 = 200º C = 473 K
From Steam Table
V1 = 0.206 m3/s
h1 = 2827.9 kJ/kg
1
1
p vn
⎛ p ⎞n
⎛ 10 ⎞1.15
∴
v 2 = 1 1 = ⎜ 1 ⎟ . v1 = ⎜ ⎟ × 0.206 = 1.5256 m3/kg
p2
⎝1 ⎠
⎝ p2 ⎠
3
∴ then steam is wet
As at 1 bar v g = 1.694 m /kg
∴
1.5256 = 0.001043 + x (1.694 – 0.001043)
∴
x = 0.9
Final temperature = 99.6º C
= u1 − u2
= (h1 − h 2 ) − (p1 v1 − p2 v 2 )
Page 134 of 265
Properties of Pure Substances
Chapter 9
[h 2 = h f 2
= (2827.9 – 2450.8) – (1000 × 0.206 – 100 × 1.5256)
= 323.7 kJ/kg
+ x h fg 2 ]
= 417.5 + 0.9 × 2257.9
= 2450.8
p v − p2 v 2
= 356.27 kJ/kg
Work done (W) = 1 1
n −1
∴
From first law of thermo dynamics
Q2 = (u2 − u1 ) + W1 – 2
= (–323.7 + 356.27) = 32.567 kJ/kg
Q9.14
Solution:
Two streams of steam, one at 2 MPa, 300°C and the other at 2 MPa, 400°C,
mix in a steady flow adiabatic process. The rates of flow of the two
streams are 3 kg/min and 2 kg/min respectively. Evaluate the final
temperature of the emerging stream, if there is no pressure drop due to
the mixing process. What would be the rate of increase in the entropy of
the universe? This stream with a negligible velocity now expands
adiabatically in a nozzle to a pressure of 1 kPa. Determine the exit
velocity of the stream and the exit area of the nozzle.
(Ans. 340°C, 0.042 kJ/K min, 1530 m/s, 53.77 cm2)
p2 = 2 MPa = 20 bar
t2 = 400º C
•
m2 = 2 kg/min
h2 = 3247.6 kJ/kg
s2 = 7.127 kJ/kg-K
1
3
p1 = 2 MPa = 20 bar
t1 = 300°C
• = 3 kg/min
m
1
2
For Steam table
h1 = 3023.5 kJ/kg
s1 = 6.766 KJ/kgK
• =m
• +m
• = 5 kg/min
m
3
1
2
p = 20 bar
3
40
s3 = 6.766 +
(6.956 – 6.766)
50
= 6.918 kJ/kg – K
For adiabatic mixing process
•
•
•
m1 h1 + m2 h 2 = m3 h 3
∴
h3 = 3113.14 kJ/kg
3113.14 − 3023.5
× 50 = 340º C
3137 − 3023.5
Rate of increase of the enthalpy of the universe
∴
•
Final temperature (t) = 300 +
•
•
•
sgen = m3 S3 − m1 S1 − m2 S2 = 0.038 kJ/K – min
After passing through nozzle if dryness fraction is x then
S3 = Sexit or 6.918 = 0.106 + x × 8.870
∴ x = 0.768
∴
he = 29.3 + 0.768 × 2484.9 = 1937.7 kJ/kg
∴
V = 2000 (3113.14 − 1937.7) = 1533.3 m/s
Page 135 of 265
Properties of Pure Substances
Chapter 9
•
Q9.15
Solution:
mv
Exit area of the nozzle =
= 0.0054 m2 = 54 cm2
V
Boiler steam at 8 bar, 250°C, reaches the engine control valve through a
pipeline at 7 bar, 200°C. It is throttled to 5 bar before expanding in the
engine to 0.1 bar, 0.9 dry. Determine per kg of steam
(a) The heat loss in the pipeline
(b) The temperature drop in passing through the throttle valve
(c) The work output of the engine
(d) The entropy change due to throttling
(e) The entropy change in passing through the engine.
(Ans. (a) 105.3 kJ/kg, (b) 5°C, (c) 499.35 kJ/kg,
(d) 0.1433 kJ/kg K, (e) 0.3657 kJ/kg K)
∴
From Steam Table
h1 = 2950.1 kJ/kg
h 2 = 2844.8
h3 = 2844.8
h 4 = hfa + xa hfga = 2345.3 kJ/kg
∴
Heat loss in pipe line = (h1 − h 2 ) = 105.3 kJ/kg
1
2
3
4
1
p1 = 8 bar
t1 = 250°C
2
p2 = 7 bar
t2 = 200°C
3
p3 = 5 bar
(b)
In throttling process h 2 = h3
∴
∴
From Steam Table
5 bar 151.8º C
hg = 2747.5
5 bar 200º C
h = 2855.4
2855.4 − 2844.8
× (200 − 151.8)
t 3 = 200 −
2855.4 − 2747.5
= 200 – 4.74 = 195.26º C
Δt = 4.74º C (drop)
(c)
Work output for the engine (W) = h3 − h 4
p4 = 0.1 bar
x4 = 0.9
t4 = 45.8°C
= (2844.8 – 2345.3) kJ/kg= 499.48 kJ/kg
(d)
∴
(e)
From Steam Table
s2 = 6.886 kJ/kg – K
(195.26 − 151.8)
(7.059 − 6.8192)
s3 = 6.8192 +
(200 − 151.8)
Δs = s3 – s2 = 0.1494 kJ/kg – K
= 7.03542 kJ/kg – K
For Steam Table
s4 = sga + 0.9 sfga = 0.649 + 0.9 × 7.501 = 7.4 kJ/kg – K
ΔS = s4 – s3 = 0.3646 kJ/kg – K
Page 136 of 265
Properties of Pure Substances
Chapter 9
Q9.16
Tank A (Figure) has a volume of 0.1 m3 and contains steam at 200°C, 10%
liquid and 90% vapour by volume, while tank B is evacuated. The valve is
then opened, and the tanks eventually come to the same pressure,
Which is found to be 4 bar. During this process, heat is transferred such
that the steam remains at 200°C. What is the volume of tank B?
(Ans. 4.89 m3)
Solution:
t1= 200°C
From Steam table
pa = 15.538 bar
V f = 1.157 – 10–3
V g = 0.12736
∴
VA = 0.1 m3
B
Initial
Initial volume of liquid =
10
× 0.1m3
100
m f = 8.643 kg
Initial mass of steam = (mg)
90
× 0.1
kg = 0.70666 kg
= 100
0.12736
∴
Total mass = 9.3497 kg
After open the valve when all over per = 4 bar at 200ºC
Then sp. Volume = 0.534 m3/kg
∴
Total volume (V) = 9.3497 × 0.534 m3 = 4.9927 m3
∴
Volume of Tank B = V – VA = 4.8927 m3
Q9.17
Calculate the amount of heat which enters or leaves 1 kg of steam
initially at 0.5 MPa and 250°C, when it undergoes the following
processes:
(a) It is confined by a piston in a cylinder and is compressed to 1 MPa
and 300°C as the piston does 200 kJ of work on the steam.
(b) It passes in steady flow through a device and leaves at 1 MPa and
300°C while, per kg of steam flowing through it, a shaft puts in 200
kJ of work. Changes in K.E. and P.E. are negligible.
It flows into an evacuated rigid container from a large source
(c)
which is maintained at the initial condition of the steam. Then 200 kJ of
shaft work is transferred to the steam, so that its final condition is 1
MPa and 300°C.
(Ans. (a) –130 kJ (b) – 109 kJ, and (c) – 367 kJ)
Page 137 of 265
Properties of Pure Substances
Chapter 9
Solution:
Initially:
t i = 250ºC
∴ From Steam Table
u i = 2729.5 kJ/kg
v i = 0.474 m3/kg
(a)
pi = 0.5 MPa = 5 bar; mass = 1 kg
h i = 2960.7 kJ/kg
After compression
p = 1 mPa = 10 bar
T = 300ºC
∴ From S.T. u = 2793.2 kJ/kg
h = 3051.2 kJ/kg
and
Winput = 200 kJ
∴ From first law of thermodynamics
Q1 – 2 = m(u2 − u1 ) + W1 – 2
[(2793.2 – 2729.5) – 200] kJ
= [63.7 – 200 kJ] = – 136.3 kJ
i.e. heat rejection to atm.
(b)
For steady flow process
V2
dQ
h1 + 1 + gz1 +
2
dm
V22
dW
+ gz2 +
= h2 +
2
dm
h or V1, V2, Z1, Z2 are negligible so
dQ
dW
= (h 2 − h1 ) +
dm
dm
= (3051.2 – 2960.7) – 200
= –109.5 kJ/kg
[heat rejection]
(c)
Energy of the gas after filling
E1 = u1 kJ/kg = 2729.5 kJ/kg
Energy of the gas after filling
E2 = u2 = 2793.2 kJ/kg
∴ ΔE = E2 – E1
= (2793.2 – 2729.5) kJ/kg
= 63.7 kJ/kg
–W
uP, hP, VP
There is a change in a volume of gas because of the collapse of the envelop to zero
volume
W1 = pi (0 – v i ) = – pi v i = – 500 × 0.474 kJ/kg = –237 kJ/kg
∴
From first law of thermodynamic
Q = ΔE + W1 + W2
= (63.7 – 237 – 200) kJ/kg = –373.3 kJ/kg
Page 138 of 265
Properties of Pure Substances
Chapter 9
Q9.18
Solution:
A sample of wet steam from a steam main flows steadily through a
partially open valve into a pipeline in which is fitted an electric coil. The
valve and the pipeline are well insulated. The steam mass flow rate is
0.008 kg/s while the coil takes 3.91 amperes at 230 volts. The main
pressure is 4 bar, and the pressure and temperature of the steam
downstream of the coil are 2 bar and 160°C respectively. Steam velocities
may be assumed to be negligible.
(a) Evaluate the quality of steam in the main.
(b) State, with reasons, whether an insulted throttling calorimeter could
be used for this test.
(Ans. (a) 0.97, (b) Not suitable)
(a)
1
2
3
2
3
1
p2 = 4 bar
3.91 × 230
kW = 0.8993 kW
1000
p3 = 2 bar; t 2 = 160ºC
t 2 = 143.6º C
h3 = 2768.8 +
•
m2 = 0.008 kg/s
•
Q = i2 R =
10
(2870.5 − 2768.8) kJ/kg
50
= 2789.14 kJ/kg
From steady flow energy equation
•
•
•
•
m h 2 + Q = m h3 + 0 : h2 = h3 −
Q
•
m
If dryness fraction of steam x then
h2 = hf2 + x hfg2
or
2676.73 = 604.7 + x × 2133
= 2676.73 kJ/kg
∴ x= 0.9714
(b) For throttling minimum enthalpy required 2686 kJ/kg if after throttling 5ºC
super heat and atm. Pressure is maintained as here enthalpy is less so it is
not possible in throttling calorimeter.
Q9.19
Solution:
Two insulated tanks, A and B, are connected by a valve. Tank A has a
volume of 0.70 m3 and contains steam at 1.5 bar, 200°C. Tank B has a
volume of 0.35 m3 and contains steam at 6 bar with a quality of 90%. The
valve is then opened, and the two tanks come to a uniform state. If there
is no heat transfer during the process, what is the final pressure?
Compute the entropy change of the universe.
(Ans. 322.6 KPa, 0.1985 kJ/K)
From Steam Table
from Steam Table
t B = 158.8º C
Sp. Enthalpy (h A ) = 2872.9 kJ/kg
3
Sp. Enthalpy (h B )
Sp. Vol (v A ) = 1.193 m /kg
Page 139 of 265
Properties of Pure Substances
Chapter 9
= (670.4 + 0.9 × 2085) = 2547 kJ/kg
A
B
VA = 0.7 m3
pA = 1.5 bar
tA = 200°C
VB = 0.35 m3
pB = 6 bar
xB = 0.9
Sp. Internal energy (u) = 2656.2 kJ/kg =
Sp. Vol. (v B ) = v Bf + x ( v Bg – v Bf ) = 0.2836 m3/kg
Sp. entropy (s) = 7.643 kJ/kg – K
Sp. in energy (uB ) = uf + x × ufg = 2376.7 kJ/kg
Sp. entropy (sB ) = 6.2748 kJ/kg – K
ufB = h fB − pfB v fB = 670.4 – 600 × 0.001101 = 669.74 kJ/kg
ufg = hfg – pfB ( v g – v f )
Q9.20
Solution:
mB =
VB
= 1.2341 kg
vB
= 1896.7 kJ/kg
V
∴ m A = A = 0.61242 kg
vA
∴
From First Law of thermodynamics
U1 = U2
∴ m A u A + mB uB = (m A + m B ) u
∴ u = 2469.4 kJ/kg
V + VB
= 0.5686 m3/kg
And sp. volume of gas after mixing = A
m A + mB
A spherical aluminum vessel has an inside diameter of 0.3 m and a 0.62
cm thick wall. The vessel contains water at 25°C with a quality of 1%.
The vessel is then heated until the water inside is saturated vapour.
Considering the vessel and water together as a system, calculate the
heat transfer during this process. The density of aluminum is 2.7 g/cm3
and its specific heat is 0.896 kJ/kg K.
(Ans. 2682.82 kJ)
4
Volume of water vapour mixture (V) = π d3i = 0.113097 m3
3
4
Ext. volume = π d 3o = 0.127709 m3
3
∴
Volume of A1 = 0.0146117 m3
∴
Mass = 39.451 kg
di = 0.3 m
0.62
do = di + 21 = 0.3 +
× 2 m = 0.3124 m
100
At 25º C; 1% quality
Page 140 of 265
Properties of Pure Substances
Chapter 9
From Steam Table
v1 = 0.001003 +
p1 = 0.0317 bar
1
(43.36) = 0.434603 m3/kg
100
1
× 24212.3 = 129.323 kJ/kg
100
u1 = h1 − p1 v1 = 127.95 kJ/kg
0.113097
kg = 0.26023 kg
Mass of water and water vapour =
0.434603
Carnot volume heating until dry saturated
So then Sp. volume vg = 0.434603 m3/kg
h1 = 104.9 +
For Steam Table
At 4.2 bar vg = 0.441
At 4.4 bar vg = 0.423
0.441 − 0.434603
(pf ) = 4.2 + 0.2 ×
= 4.27 bar
0.441 − 0.423
0.07
h f = 2739.8 +
(2741.9 − 2739.8) = 2740.55 kJ/kg
Then
0.2
t f = 146º C
uf = h f − pf v f = 2555 kJ/kg
∴
Heat required to water = m(uf − u1 )
= 0.26023(2555 – 127.95) kJ
= 631.58 kJ
Heat required for A1
= 39.451 × 0.896 × (146 – 25)
= 4277.2 kJ
Total heat required = 4908.76 kJ
Q9.21
Solution:
Steam at 10 bar, 250°C flowing with negligible velocity at the rate of 3
kg/min mixes adiabatically with steam at 10 bar, 0.75 quality, flowing
also with negligible velocity at the rate of 5 kg/min. The combined
stream of steam is throttled to 5 bar and then expanded isentropically in
a nozzle to 2 bar. Determine
(a) The state of steam after mixing
(b) The state of steam after throttling
(c) The increase in entropy due to throttling
(d) The velocity of steam at the exit from the nozzle
(e) The exit area of the nozzle. Neglect the K.E. of steam at the inlet to
the nozzle.
(Ans. (a) 10 bar, 0.975 dry, (b) 5 bar, 0.894 dry, (c) 0.2669 kJ/kg K,
(d) 540 m/s, (e) 1.864 cm2)
From Steam Table
h1 = 2942.6 kJ/kg
h 2 = 762.6 + 0.75 × 2013.6 = 2272.8 kJ/kg
3 × 2942.6 + 5 × 2272.8
∴
h3 =
8
= 2524 kJ/kg
Page 141 of 265
Properties of Pure Substances
Chapter 9
•
m 1 = 3 kg/min
1
10 bar
250°C
3
4
1
5
5 bar
1
10 bar
x2 = 0.75
• = 5 kg/min 1
m
3
4
p3 = 10 bar
• = 8 kg/min
m
3
t3 = 180°c
2
h3 = 762.6 + x 3 × 2013.6 or x 3 = 0.87474
(b)
h 4 = 2524 kJ/kg = 640.1 + x 4 × 2107.4
x 4 = 0.89395
s4 = 1.8604 + x 4 × 4.9588 = 6.2933 kJ/kg – K
s5 = s4
∴
5
t4 = 151.8°c
(a)
∴
2 bar
at 2 bar quality of steam
6.2933 = 1.5301 + x 5 × 5.5967
x 5 = 0.851
∴
(c)
h5 = 504.7 + 0.851 × 2201.6 = 2378.4 kJ/kg
s3 = sf + x 3 s fg = 2.1382 + 0.89395 × 4.4446 = 6.111451 kJ/kg
Δs = s4 – s3 = 6.2933 – 6.11145 = 0.18185 kJ/kg – K
(d)
V=
(e)
A=
⇒
Q9.22
Solution:
2000(2524 − 2378.4) = 540 m/s
mv
V
m × x 5 . 0.885
8
0.885 2
=
× 0.851 ×
m = 1.86 cm2
V
60
540
Steam of 65 bar, 400°C leaves the boiler to enter a steam turbine fitted
with a throttle governor. At a reduced load, as the governor takes action,
the pressure of steam is reduced to 59 bar by throttling before it is
admitted to the turbine. Evaluate the availabilities of steam before and
after the throttling process and the irreversibility due to it.
(Ans. I = 21 kJ/kg)
From Steam Table
h1 = 3167.65 kJ/kg
h 2 = 3167.65 kJ/kg
s1 = 6.4945 kJ/kg-K
t 2 = 396.6º C
Page 142 of 265
Properties of Pure Substances
Chapter 9
p1 = 65 bar
h1 = 400°C
1
2
1
2
60 bar
46.6
(6.541 − 6.333)
50
= 6.526856
– s3 = 0.032356 kJ/kg – K
s2 = 6.333 +
∴ Δs = s4
Atmospheric Pressure
p0 = 1 bar
T0 = 25º C
∴ Availability before throttling
V2
ψ = (h1 − h 0 ) – T0 ( s1 – s0 ) + 1 + g (2 Z0)
2
Same as example 9.14
Q9.23
A mass of wet steam at temperature 165°C is expanded at constant
quality 0.8 to pressure 3 bar. It is then heated at constant pressure to a
degree of superheat of 66.5°C. Find the enthalpy and entropy changes
during expansion and during heating. Draw the T–s and h–s diagrams.
(Ans. – 59 kJ/kg, 0.163 kJ/kg K during expansion and 676 kJ/kg,
1.588 kJ/kg K during heating)
Solution:
p1 = 7 bar
t1 = 165º C
For Steam Table
h1 = h f + 0.8 h fg
s1
= 2349 kJ/kg
= sf + 0.8 × sfg
= 5.76252 kJ/kg – K
For Steam Table at 3 bar
1
165°C
T
ar
3
66 .5°C
x = 0.8
2
b
3
t 2 = 133.5º C
h 2 = 561.4 + 0.8 × 2163.2 = 2292 kJ
s2 = 1.6716 + 0.8 × 5.3193
S
= 5.92704 kJ/kg – K
∴
temperature of (3)
t3 = 200ºC
∴
h3 = 2865.6 kJ/K
s3 = 7.311 kJ/kg-K
∴
Enthalpy charge in expansion = (h1 − h 2 ) = 57 kJ/kg
Entropy charge in expansion = ( s2 – s1 ) = 0.16452 kJ/kg-K
Enthalpy charge in heating = h3 − h 2 = 573.6 kJ/kg
Entropy charge in heating = s3 − s2 = 1.38396 kJ/kg – K
Page 143 of 265
Page 144 of 265
Properties of Gases and Gas Mixtures
Chapter 10
10. Properties of Gases and Gas Mixture
Some Important Notes
1.
As p → 0, or T → ∞, the real gas approaches the ideal gas behaviour.
R = 8.3143 kJ/kmole-K
2.
Tds = du + pdv
Tds = dh – vdp
2
γ =1 +
N
For mono-atomic gas N = 3
For di -atomic gas N = 5
For Tri-atomic gas N = 6
3.
4.
[N = degrees of freedom]
Reversible adiabatic process
T
⎛p ⎞
pv γ = C ; 2 = ⎜ 2 ⎟
T1 ⎝ p1 ⎠
5.
γ −1
γ
=
⎛ v1 ⎞
⎜v ⎟
⎝ 2⎠
γ −1
For isentropic process
RT1
u2 − u1 =
γ −1
(i)
For closed system
γ −1
γ −1
⎡
⎤
⎡
⎤
γ
γ
p
p
γ
⎛
⎞
⎛
⎞
2
⎢
⎢ 2
⎥
(RT1 ) ⎜ ⎟
− 1⎥
− 1 ; h 2 − h1 =
⎢⎣⎝ p1 ⎠
⎥⎦
⎢⎣⎜⎝ p1 ⎟⎠
⎥⎦
γ −1
2
p v −p v
∫1 pd v = 1 1γ − 12 2
2
6.
γ
(p1 v1 − p2 v 2 )
1
γ
−
1
Isobaric process (p = C), n = 0, pvº = C
Isothermal process (T = C), n = 1, pv1 = RT
Isentropic process (s = C), n = γ , pvγ = C
Isometric or isobaric process (V = C), n = ∞
7.
For minimum work in multistage compressor, p2 =
For steady flow
∫ v dp
=
p2 p3
=
p1 p2
(ii) Equal discharge temperature (T2 = T3 )
(i)
Equal pressure ratio i.e.
Page 145 of 265
p1 p3
Properties of Gases and Gas Mixtures
Chapter 10
p3
p2
p1
T
2
3
2′
And (iii) Equal work for the two stages.
1
S
1
8.
9.
⎛ p ⎞n
Volumetric Efficiency ( ηvol ) = 1 + C − C ⎜ 2 ⎟
⎝ p1 ⎠
Clearance volume
Where, C =
Piston displacement volume
Equation of states for real gas
(i)
a ⎞
⎛
Van der Waals equation: ⎜ p + 2 ⎟ (v − b) = RT
v ⎠
⎝
RT
a
−
p=
or
v − b v2
or
(ii)
3 ⎞
⎛
⎜ pr + v 2 ⎟ (3 v r − 1) = 8Tr
r ⎠
⎝
Beattie Bridge man equation
RT (1 − e)
A
(v + B) − 2
2
v
v
a
b⎞
C
⎛
⎞
⎛
A = A 0 ⎜ 1 − ⎟ ; B = B0 ⎜ 1 − ⎟ ; e =
v⎠
vT3
v⎠
⎝
⎝
p=
Where
‘Does not’ give satisfactory results in the critical point region.
(iii)
Virial Expansions:
pv
= 1 + B′p + C′p2 + D′p3 + ……………
RT
pv
B C
D
Or
= 1 + + 2 + 3 + ....... α
v v
v
RT
Page 146 of 265
Properties of Gases and Gas Mixtures
Chapter 10
a = 3 pc v 2c ; b =
10.
vc
8 pc v c
; values of Z at critical point 0.375 for Van der Waal
; R=
3
3 Tc
gas.
a
bR
μ = x1 μ1 + x 2 μ2 + ....... + x c μ c
Boyle temperature (TB) =
11.
12.
m1R1 + m2 R 2 + ....... mc R c
m1 + m2 + ........... mc
m u + m2 u2 + ....... m c uc
um = 1 1
;
m1 + m2 + ........... m c
Rm =
c pm
=
m1 cP1 + m2 cP2 + ....... mc cPc
m1 + m2 + ........... mc
hm =
m1h1 + m2 h 2 + ....... m c h c
m1 + m 2 + ........... m c
c vm
=
m1 cv1 + m2 cv2 + ....... mc cv c
m1 + m2 + ........... mc
p ⎤
p
p
⎡
sf − si = − ⎢m1R1 ln 1 + m2 R 2 ln 2 + ...... + m c R c ln c ⎥
p
p
p⎦
⎣
Gibbs function G = RT ∑ n x ( φk + ln p + ln x k )
13.
Questions with Solution P. K. Nag
Q.10·1
Solution:
What is the mass of air contained in a room 6 m × 9 m × 4 m if the
pressure is 101.325 kPa and the temperature is 25°C?
(Ans. 256 kg)
Given pressure (p) = 101.325 kPa
Temperature (T)
= 25ºC = (25 + 273) K = 298 K
Volume (V)
= 6 × 9 × 4 m3 = 216 m3
From equation of states
pV = mRT
For air R = 0.287 kJ/kg – K, Gas constant mass is m kg
∴
Q.10.2
(c)
Solution:
m=
pV
RT
=
101.325 × 216
kg = 255.9 kg
0.287 × 298
The usual cooking gas (mostly methane) cylinder is about 25 cm in
diameter and 80 cm in height. It is changed to 12 MPa at room
temperature (27°C).
(a) Assuming the ideal gas law, find the mass of gas filled in the
cylinder.
(b) Explain how the actual cylinder contains nearly 15 kg of gas.
If the cylinder is to be protected against excessive pressure by means of
a fusible plug, at what temperature should the plug melt to limit the
maximum pressure to 15 MPa?
Given diameter
Height
(D) = 25 cm = 0.25 m
(H) = 80 cm = 0.8 m
π D2
Volume of the cylinder
∴
(V) =
× H = 0.03927 m3
4
Gas pressure
(p) = 12 MPa = 12000 kPa
Page 147 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Temperature
(a)
(T) = 27º C = 300 K
Mass of gas filled in the cylinder
⎡
⎤
pV
R 8.3143
=
kJ/kg – K = 0.51964 ⎥
⎢ Here R = Gas constant =
RT
M
16
⎣
⎦
= 3.023 kg
m=
(b)
In cooking gas main component is ethen and it filled in 18 bar pressure. At
that pressure it is not a gas it is liquid form in atmospheric temperature so
its weight is amount 14 kg.
(c)
Let temperature be T K, then pressure, p = 15 MPa = 15000 kPa
∴
Q.10.3
T=
pV
15000 × 0.03927
=
= 375 K = 102º C
mR
3.023 × 0.51964
A certain gas has cP = 0.913 and cV = 0.653 kJ/kg K. Find the molecular
weight and the gas constant R of the gas.
Solution:
Gas constant, R = c p – c v = (0.913 – 653) kJ/kg – K = 0.26 kJ/kg – K
If molecular weight,( M )kJ/kg – mole
R
8.3143
=
kJ/kg – mole = 31.98 kJ/kg – mole
Then R = MR
∴M=
R
0.26
Q.10.4
From an experimental determination the specific heat ratio for
acetylene (C2H2 ) is found to 1.26. Find the two specific heats.
Solution:
Gas constant of acetylene (C2 H2 ) (R) =
R
8.3143
=
kJ/kg – K = 0.3198 kJ/kg – K
M
26
As adiabatic index ( γ ) = 1.26 then
γ
cp =
R = 1.55 kJ/kg – K
γ −1
R
cv =
= 1.23 kJ/kg – K
and
γ −1
Q.10.5
Solution:
Find the molal specific heats of monatomic, diatomic, and polyatomic
gases, if their specific heat ratios are respectively 5/3, 7/5 and 4/3.
γ
Mono-atomic: c p =
R = 20.79 kJ/kg – mole – K;
γ −1
R
= 12.47 kJ/kg – mole – K
γ −1
γ
cp =
R = 29.1 kJ/kg – mole – K;
γ −1
cv =
Di-atomic:
cv =
R
= 20.79 kJ/kg – mole – K
γ −1
Page 148 of 265
Properties of Gases and Gas Mixtures
Chapter 10
cp =
Polyatomic:
γ
R = 33.26 kJ/kg – mole – K;
γ −1
cv = 24.94 kJ/kg – mole – K
Q.10.6
Solution:
A supply of natural gas is required on a site 800 m above storage level.
The gas at - 150°C, 1.1 bar from storage is pumped steadily to a point on
the site where its pressure is 1.2 bar, its temperature 15°C, and its flow
rate 1000 m 3 /hr. If the work transfer to the gas at the pump is 15 kW,
find the heat transfer to the gas between the two points. Neglect the
change in K.E. and assume that the gas has the properties of methane (C
H4 ) which may be treated as an ideal gas having γ = 1.33 (g = 9.75 m/ s2 ).
(Ans. 63.9 kW)
Given:
At storage
(p1 ) = 1.1 bar = 110 kPa
(T1 ) = –150ºC = 123 K
p3 = 1.2 bar = 120 kPa
T3 = 288 K
•
(V 3 ) = 1000 m3/m =
Flow rate
Gas constant (R) =
5 3
m /s
18
R
= 0.51964 kJ/kg – K
16
∴
•
•
•
∴ p3 V3 = m RT3
p V3
∴ m =
= 0.22273 kg/s
RT3
2
•
⎛ dW ⎞
Pump work ⎜
⎟ = –15 kW
⎝ dt ⎠
∴
From steady flow energy equation
dQ
dW
•
•
m(h1 + 0 + gZ1 ) +
= m( h3 + 0 + g Z3 ) +
dt
dt
(Z
−
Z
)
dW
dQ
• ⎡
1 ⎤
∴
= m ⎢(h3 − h1 ) + g 3
+
dt
1000 ⎥⎦
dt
⎣
Δ Z ⎤ dW
• ⎡
= m ⎢c P (T3 − T1 ) + g
+
1000 ⎥⎦
dt
⎣
9.75 × 800 ⎤
⎡
= 0.22273 ⎢2.0943 × (288 − 123) +
1000 ⎥⎦
⎣
+ ( −15)
= 63.7 kJ/s = 63.7 kW (heat given to the system)
Q.10.7
3
γ
cp =
R = 2.0943 kJ/kg
γ −1
800m
P
1
A constant volume chamber of 0.3 m 3 capacity contains 1 kg of air at
5°C. Heat is transferred to the air until the temperature is 100°C. Find
the work done, the heat transferred, and the changes in internal energy,
enthalpy and entropy.
Page 149 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
Constant volume (V) = 0.3 m3
T2 = 100ºC = 373 K
p
p2 = 1 × T2 = 357 kPa
T1
∴
Mass (m) = 1 kg
T1 = 5º C = 278 K
∴
mRT1
= 265.95 kPa
V
Work done = ∫ pdV = 0
p1 =
∫ du + ∫ dW = ∫ dW = m c ∫ dT = m c
Change in internal Energy = ∫ du = 68.21 kJ
Change in Enthalpy = ∫ dh = m c (T – T ) = 95.475 kJ
Heat transferred Q =
v
P
Change in Entropy =
∫ d s = s2 –
2
v
(T2 – T1 ) = 68.21 kJ
1
s1 = m c p ln
V2
p
+ m c v ln 2
V1
p1
p2
357
= 1 × 0.718 × ln
265.95
p1
= 0.2114 kJ/kg – K
= m c v ln
Q.10.8
One kg of air in a closed system, initially at 5°C and occupying 0.3 m3
volume, undergoes a constant pressure heating process to 100°C. There
is no work other than pdv work. Find (a) the work done during the
process, (b) the heat transferred, and (c) the entropy change of the gas.
Solution:
T1 = 278 K
V1 = 0.3 m3
m = 1 kg
∴ p1 = 265.95 kPa
T2 = 100º C = 373 K
p2 = 265.95 kPa
mRT2
∴ V2 =
= 0.40252 m3
p2
(a) Work during the process
2
(W12) =
∫ p dV
= p(V2 − V1 ) = 27.266 kJ
p
2
1
1
(b) Heat transferred Q1 – 2 = u2 – u1 + W12
= mc v (T2 – T1 ) + W1 – 2 = 95.476 kJ
V
(c) Entropy change of the gas
V
p
s2 – s1 = mc p ln 2 + mc v ln 2
V1
p1
v
= m c p ln 2 = 0.29543 kJ/kg – K
v1
Q.10.9
0.1 m 3 of hydrogen initially at 1.2 MPa, 200°C undergoes a reversible
isothermal expansion to 0.1 MPa. Find (a) the work done during the
process, (b) the heat transferred, and (c) the entropy change of the gas.
Page 150 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
V1 = 0.1 m3
p1 = 1.2 MPa = 1200 kPa
T1 = 473 K
1
1
p
2
T
2
S
V
R
8.3143
=
kJ/kg – K = 4.157 kJ/kg – K
M
2
p V
m = 1 1 = 0.06103 kg
RT1
Reversible isothermal expansion
So
T2 = T1 = 473 K
Enthalpy change (Δh) = m c p (T2 – T1 ) = 0
R=
And
p2 = 0.1 MPa = 100 kPa
Heat transferred (Q) = Δu + ΔW
mRT2
3
∴
V2 =
= 1.2 m
p2
2
= u2 − u1 + ∫ p dV
1
dV
= 0 + RT ∫
V
⎛V ⎞
= RT ln ⎜ 2 ⎟
⎝ V1 ⎠
pV = RT
∴
p=
RT
V
⎛ 1.2 ⎞
= 4.157 × 473 × ln ⎜
⎟
⎝ 0.1 ⎠
= 4886 kJ
2
Work done (W) =
∫ p dV
= 4886 kJ
1
Entropy change, s2 – s1 = mc p ln
V2
p
+ mc v ln 2
V1
p1
⎡
⎛ 1.2 ⎞
⎛ 100 ⎞ ⎤
= 0.06103 ⎢14.55 ln ⎜
⎟ + 10.4 ln ⎜
⎟⎥
⎝ 0.1 ⎠
⎝ 1200 ⎠ ⎦
⎣
= 0.6294 kJ/kg – K
For H2 diatomic gas (γ = 1.4)
γ
R
= 10.4 kJ/kg – K
cp =
R = 14.55 kJ/kg – K; cv =
γ −1
γ −1
Q.10.10
Air in a closed stationary system expands in a reversible adiabatic
process from 0.5 MPa, 15°C to 0.2 MPa. Find the final temperature, and
per kg of air, the change in enthalpy, the heat transferred, and the work
done.
Page 151 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
p1 = 0.5 MPa = 500 kPa
T1 = 15ºC = 288 K
Let mass is 1 kg
∴
v1 = 1 × R × T1 = 0.1653 m3/kg
p1
p2 = 0.2 MPa = 200 kPa
∴
p1 v1γ = p2 v 2γ :
1
1
γ
⎛p ⎞
v2 = v1 × ⎜ 1 ⎟ = 0.31809 m3/kg
⎝ p2 ⎠
γ −1
T
γ −1
T1
⎛p ⎞ γ
⎛p ⎞ γ
∴
∴ T2 = T1 × ⎜ 2 ⎟
= ⎜ 1⎟
T2
⎝ p2 ⎠
⎝ p1 ⎠
= 222 K
Change of Enthalpy
(ΔH) = mc p (T2 – T1 ) = –66.33 kJ/kg
2
S
The Heat transferred (Q) = 0
The work done
2
p v − p2 v 2
(W) = ∫ p d v = 1 1
γ −1
1
= 47.58 kJ/kg
Q.10.11
Solution:
If the above process occurs in an open steady flow system, find the final
temperature, and per kg of air, the change in internal energy, the heat
transferred, and the shaft work. Neglect velocity and elevation changes.
T
⎛p ⎞
Final temperature will be same because then also 2 = ⎜ 2 ⎟
T1
⎝ p1 ⎠
i.e. T2 = 222 K
Change in internal energy = Δu = mc v (T2 – T1 ) = –47.4 kJ/kg
2
(W) = − ∫ v dp =
Shaft work
1
γ −1
γ
valid.
γ
[p1 v1 − p2 v 2 ] = +66.33 kJ/kg
γ −1
Heat transferred:
h1 + 0 + 0 +
∴
Q.10.12
dQ
dW
= h2 + 0 + 0 +
dm
dm
dQ
dW
= (h 2 − h1 ) +
= –66.33 + 66.33 = 0
dm
dm
[As it is reversible adiabatic so dQ = 0]
The indicator diagram for a certain water-cooled cylinder and piston air
compressor shows that during compression pv1.3 = constant. The
compression starts at 100 kPa, 25°C and ends at 600 kPa. If the process is
reversible, how much heat is transferred per kg of air?
Page 152 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
p1 = 100 kPa
T1 = 298 K
RT1
= 0.8553 m3/kg
∴ v1 =
p1
p2 = 600 kPa
1
n
⎛p ⎞
v 2 = v1 ⎜ 1 ⎟ = 0.21554 m3/kg
⎝ p2 ⎠
2
pV
1.3
= constant
p
1
n −1
n
∴
⎛p ⎞
= 451 K
T2 = T1 × ⎜ 2 ⎟
⎝ p1 ⎠
dQ
dW
h1 + 0 + 0 +
= h2 + 0 + 0 +
dm
dm
dQ
dW
= (h 2 − h1 ) +
dm
dm
n[p1 v1 − p2 v 2 ]
dW
=
n −1
dm
= –189.774 kJ/kg
= Cp(T2 – T1) -189.774
= 153.765 – 189.774
= –36 kJ/kg
[Heat have to be rejected]
∴
Q.10.13
V
An ideal gas of molecular weight 30 and γ = 1.3 occupies a volume of 1.5
m 3 at 100 kPa and 77°C. The gas is compressed according to the law
pv1.25 = constant to a pressure of 3 MPa. Calculate the volume and
temperature at the end of compression and heating, work done, heat
transferred, and the total change of entropy.
Solution:
R
= 0.27714 kJ/kg – K
30
γ = 1.3; n = 1.25
R
= 0.9238 kJ/kg – K
∴ cv =
γ −1
R
cP = γ
= 1.2 kJ/kg – K
γ −1
p1 = 100 kPa; V1 = 1.5 m3; T1 = 350 K
p2 = 3 MPa = 3000 kPa
R=
2
p
1
⎛ p ⎞n
V2 = V1 ⎜ 1 ⎟ = 0.09872 m3
⎝ p2 ⎠
pV
m = 1 1 = 1.5464 kg
RT1
pV
∴ T2 = 2 2 = 691 K
mR
∴
Page 153 of 265
1
V
Properties of Gases and Gas Mixtures
Chapter 10
2
Work done (W1 – 2) =
∴ p1 V
n
1
∫ pdV
1
n
2
n
= p V = p2 V
= p1 V1n
2
p1 V1n
dV
=
∫1 V n − n + 1
1 ⎤
⎡ 1
⎢ Vn − 1 − Vn − 1 ⎥
1
⎣ 2
⎦
p2 V2 − p1 V1
p V − p2 V2
= 1 1
1−n
n −1
100 × 1.5 − 3000 × 0.09872
kJ = –584.64 kJ
=
1.25 − 1
=
Heat transfer Q = u2 – u1 + W1 – 2
= mc v (T2 – T1 ) + W1 – 2
= [1.5464 × 0.9238 (691 – 350) – 584.64] kJ
= – 97.5 kJ
p
V ⎤
⎡
ΔS = S2 – S1 = ⎢mc v ln 2 + mc P ln 2 ⎥
p1
V1 ⎦
⎣
= – 0.19 kJ/K
Q.10.14
Calculate the change of entropy when 1 kg of air changes from a
temperature of 330 K and a volume of 0.15 m 3 to a temperature of 550 K
and a volume of 0.6 m 3 .
If the air expands according to the law, pv n = constant, between the
same end states, calculate the heat given to, or extracted from, the air
during the expansion, and show that it is approximately equal to the
change of entropy multiplied by the mean absolute temperature.
Solution:
Try please.
Q.10.15
0.5 kg of air, initially at 25°C, is heated reversibly at constant pressure
until the volume is doubled, and is then heated reversibly at constant
volume until the pressure is doubled. For the total path, find the work
transfer, the heat transfer, and the change of entropy.
Solution:
Try please.
Q.10.16
An ideal gas cycle of three processes uses Argon (Mol. wt. 40) as a
working substance. Process 1-2 is a reversible adiabatic expansion from
0.014 m 3 , 700 kPa, 280°C to 0.056 m 3 . Process 2-3 is a reversible
isothermal process. Process 3-1 is a constant pressure process in which
heat transfer is zero. Sketch the cycle in the p-v and T-s planes, and find
(a) the work transfer in process 1-2, (b) the work transfer in process 2-3,
and (c) the net work of the cycle. Take γ = 1.67.
Solution:
Try please.
Q.10.17
A gas occupies 0.024 m 3 at 700 kPa and 95°C. It is expanded in the nonflow process according to the law pv1.2 = constant to a pressure of 70 kPa
Page 154 of 265
Properties of Gases and Gas Mixtures
Chapter 10
after which it is heated at constant pressure back to its original
temperature. Sketch the process on the p-v and T-s diagrams, and
calculate for the whole process the work done, the heat transferred, and
the change of entropy. Take c p = 1.047 and c V = 0.775 kJ/kg K for the gas.
Solution:
Try please.
Q.10.18
0.5 kg of air at 600 kPa receives an addition of heat at constant volume
so that its temperature rises from 110°C to 650°C. It then expands in a
cylinder poly tropically to its original temperature and the index of
expansion is 1.32. Finally, it is compressed isothermally to its original
volume. Calculate (a) the change of entropy during each of the three
stages, (b) the pressures at the end of constant volume heat addition and
at the end of expansion. Sketch the processes on the p-v and T-s
diagrams.
Try please.
Solution:
Q.10.19
0.5 kg of helium and 0.5 kg of nitrogen are mixed at 20°C and at a total
pressure of 100 kPa. Find (a) the volume of the mixture, (b) the partial
volumes of the components, (c) the partial pressures of the components,
(d) the mole fractions of the components, (e) the specific heats cP and cV
of the mixture, and (f) the gas constant of the mixture.
Solution:
Try please.
Q.10.20
A gaseous mixture consists of 1 kg of oxygen and 2 kg of nitrogen at a
pressure of 150 kPa and a temperature of 20°C. Determine the changes in
internal energy, enthalpy and entropy of the mixture when the mixture
is heated to a temperature of 100°C (a) at constant volume, and (b) at
constant pressure.
Solution:
Try please.
Q.10.21
A closed rigid cylinder is divided by a diaphragm into two equal
compartments, each of volume 0.1 m 3 . Each compartment contains air at
a temperature of 20°C. The pressure in one compartment is 2.5 MPa and
in the other compartment is 1 MPa. The diaphragm is ruptured so that
the air in both the compartments mixes to bring the pressure to a
uniform value throughout the cylinder which is insulated. Find the net
change of entropy for the mixing process.
Solution:
Try please.
Q.10.22
A vessel is divided into three compartments (a), (b), and (c) by two
partitions. Part (a) contains oxygen and has a volume of 0.1 m 3 , (b) has a
volume of 0.2 m 3 and contains nitrogen, while (c) is 0.05 m 3 and holds
C O2 . All three parts are at a pressure of 2 bar and a temperature of 13°C.
When the partitions are removed and the gases mix, determine the
change of entropy of each constituent, the final pressure in the vessel
and the partial pressure of each gas. The vessel may be taken as being
completely isolated from its surroundings.
(Ans. 0.0875, 0.0783,
0.0680
kJ/K; 2 bar; 0.5714, 1.1429, 0.2857 bar.)
Page 155
of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
a
b
c
0.1 m 3
0.2 m 3
0.05 m 3
N2
O2
W2
p = 2 bar = 200 kPa
T = B° C = 286 K
After mixing temperature of the mixture will be same as before 13ºC = 286 K and
also pressure will be same as before 2 bar = 200 kPa. But total volume will be
V = Va + Vb + Vc
= (0.1 + 0.2 + 0.05) = 0.35 m3
pVa
200 × 0.1
ma =
=
∴
kg = 0.26915 kg
8.3143
Ra T
× 286
32
pVa
200 × 0.2
mb =
=
kg = 0.471 kg
8.3143
RbT
× 286
28
pVc
200 × 0.05
mc =
=
kg = 0.18504 kg
8.319
Rc T
× 286
44
v ⎤
T
p
⎡ p
∴
ΔS = S2 – S1 = mc P ln 2 − mR ln 2
Here T2 = T1 so ⎢∵ 2 = 1 ⎥
T1
p1
⎣ p1 v 2 ⎦
(S2 − S1 )O2 = mO2 R O2 ln
Vo
8.3143
⎛ 9.35 ⎞
× ln ⎜
= 0.26915 ×
⎟
VO2
32
⎝ 0.1 ⎠
= 0.087604 kJ/K
8.3143
⎛ V ⎞
⎛ 0.35 ⎞
= 0.471 ×
× ln ⎜
(S2 − S1 )N2 = mN2 R N2 ln ⎜
⎟ = 0.078267 kJ/K
⎟
32
⎝ 0.2 ⎠
⎝ Vn2 ⎠
8.3143
⎛ V ⎞
⎛ 0.35 ⎞
= 0.18504 ×
× ln ⎜
(S2 − S1 )CO2 = mCO2 RCO2 ln ⎜
⎟ = 0.06804 kJ/K
⎟
44
⎝ 0.05 ⎠
⎝ VCO2 ⎠
Partial pressure after mixing
0.1
Mole fraction of
O2 (x O2 ) =
0.35
0.2
Mole fraction of
N 2 (x N2 ) =
0.35
0.05
Mole fraction of
CO2, x O2 =
0.35
[∵ At same temperature and pressure same mole of gas has same]
0.1
× 200 = 57.143 kPa
O2 ; (pO2 ) = x O2 × p =
∴ Partial pressure of
0.35
0.2
× 200 = 114.29 kPa
Partial pressure of
N 2 ; (pN2 ) = x N 2 × p =
0.35
0.05
Partial pressure of
× 200 = 28.514 kPa
CO2 ; (pCO2 ) = x CO2 × p =
0.35
( )
Page 156 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Q.10.23
A Carnot cycle uses 1 kg of air as the working fluid. The maximum and
minimum temperatures of the cycle are 600 K and 300 K. The maximum
pressure of the cycle is 1 MPa and the volume of the gas doubles during
the isothermal heating process. Show by calculation of net work and
heat supplied that the efficiency is the maximum possible for the given
maximum and minimum temperatures.
Solution:
Try please.
Q.10.24
An ideal gas cycle consists of three reversible processes in the following
sequence: (a) constant volume pressure rise, (b) isentropic expansion to
r times the initial volume, and (c) constant pressure decrease in volume.
Sketch the cycle on the p-v and T'-s diagrams. Show that the efficiency of
the cycle is
r γ − 1 − γ ( r − 1)
ηcycle =
rγ − 1
4
Evaluate the cycle efficiency when y =
and r = 8.
3
(Ans. ( η = 0.378))
For process 1 – 2 constant volume heating
Q1 – 2 = Δu + pdv
= mc v ΔT + pdv
Solution:
= mc v ΔT = mc v (T2 – T1 )
2
2
T
1
p
3
3
1
V
S
Q2 – 3 = 0 as isentropic expansion.
Q3 – 1 = mc P ΔT = mc P (T3 – T1 )
∴
Efficiency = 1 −
heat rejection
heat addition
⎛ T3
⎞
− 1⎟
⎜
mc p (T3 − T1 )
T
⎠
= 1−γ⎝ 1
= 1−
mc v (T2 − T1 )
⎛ T2
⎞
⎜ T − 1⎟
⎝ 1
⎠
p1 v1
p2 v 2
T2
p
=
as V1 = V2
= 2 = rγ
∴
Here
T1
T2
T1
p1
γ
And p2 v 2γ = p3 v 3γ
And
or
p3 v 3
pv
= 2 2
T1
T3
Page 157 of 265
p2
⎛v ⎞
= ⎜ 3 ⎟ = rγ
p3
⎝ v2 ⎠
p2
= rγ
p1
as p3 = p1 then
Properties of Gases and Gas Mixtures
Chapter 10
or
If
T3
v
v
= 3 = 3 =r
T1
v1
v2
γ=
∴η= 1−
γ( γ − 1)
r γ − 1 − γ(r − 1)
Proved
=
rγ − 1
rγ − 1
4
and r = 8 then
3
ηcycle =
r γ − 1 − γ(r − 1)
rγ − 1
4
(8 − 1)
3
η= 1− 4
= 0.37778
3
(8 − 1)
Q.10.25
Solution :
Using the Dietetic equation of state
⎛
RT
a ⎞
P=
.exp ⎜ −
⎟
v−b
⎝ RTv ⎠
(a) Show that
a
a
pc = 2 2 , v c = 2b, Tc =
4Rb
4e b
(b) Expand in the form
B C
⎛
⎞
pv = RT ⎜ 1 + + 2 + .... ⎟
v v
⎝
⎠
(c) Show that
a
TB =
bR
Try please.
Q.10.26
The number of moles, the pressures, and the temperatures of gases a, b,
and c are given below
Gas
m (kg mol)
P (kPa)
t (0C)
N2
1
350
100
CO
3
420
200
O2
2
700
300
If the containers are connected, allowing the gases to mix freely, find (a)
the pressure and temperature of the resulting mixture at equilibrium,
and (b) the change of entropy of each constituent and that of the
mixture.
Solution :
Try please.
Q.10.27
Calculate the volume of 2.5 kg moles of steam at 236.4 atm. and 776.76 K
with the help of compressibility factor versus reduced pressure graph.
At this volume and the given pressure, what would the temperature be
in K, if steam behaved like a van der Waals gas?
The critical pressure, volume, and temperature of steam are 218.2 atm.,
57 cm 3 /g mole, and 647.3 K respectively.
Solution :
Try please.
Q.10.28
Two vessels, A and B, each of volume 3 m 3 may be connected together by
a tube of negligible volume. Vessel a contains air at 7 bar, 95°C while B
Page 158 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
contains air at 3.5 bar, 205°C. Find the change of entropy when A is
connected to B. Assume the mixing to be complete and adiabatic.
(Ans. 0.975 kJ/kg K)
VA = VB = 3m3
pA = 7 bar = 700 kPa
TA = 95ºC = 368 K
pB = 3.5 bar = 350 kPa
TB = 205ºC = 478 K
∴
mA =
pA VA
= 19.883 kg
RTA
mB =
In case of Adiabatic mixing for closed system
Internal energy remains constant.
∴
UA + UB = U
m A c v . TA + m B c v . TB = (m A + m B ) c v T
or
m A TA + mB TB
= 398.6 K
m A + mB
After mixing partial for of A
⎡ Total pressure
⎤
⎢
⎥
⎢∴ p = mRT = 525.03 kPa ⎥
V
⎣⎢
⎦⎥
m RT
= 379.1 kPa
pAf = A
V
m RT
pBf = B
= 145.93 kPa
V
p
T
ΔSA = SAf – SA = m A c p ln
− m A R ln Af
TA
pA
= 5.0957 kJ/K
p
T
− m B R ln Bf
ΔsBf − sB = m B c P ln
TB
pB
= 0.52435 kJ/kg
∴
ΔSuniv = ΔSA + ΔSB = 5.62 kJ/K
∴
Q.10.29
T=
pB VB
= 7.6538 kg
RTB
A
B
3m 3
3m 3
pA = 7 bar
TA = 95°C
pB = 3.5 bar
TB = 205°C
An ideal gas at temperature T1 is heated at constant pressure to T2
n
and
then expanded reversibly, according to the law pv = constant, until the
temperature is once again T1 What is the required value of n if the
changes of entropy during the separate processes are equal?
⎛
⎛
2γ ⎞ ⎞
⎜ Ans. ⎜ n =
⎟⎟
γ + 1⎠⎠
⎝
⎝
Page 159 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Solution:
Let us mass of gas is 1 kg
2
p
T2
T
1
T1
1
T2
2
3 (T1 )
3
S
Then
V
v2
p
T
T
p ⎤
T
γR
⎡
× ln 2
+ c v ln 2 = ⎢c p ln 2 − R ln 2 ⎥ = c p ln 2 =
T1
v1
p1
T1
p1 ⎦
T1
γ −1
⎣
T
p
= c p ln 3 − R ln 3
T2
p2
s2 – s1 = 1 × c P ln
or
s3 – s 2
n −1
n −1
T
⎛p ⎞ ⎛T ⎞ n
⎛p ⎞ n
Hence T3 = T1 and 3 = ⎜ 3 ⎟
∴ ⎜ 3⎟ = ⎜ 3⎟
T2
⎝ p2 ⎠ ⎝ T2 ⎠
⎝ p2 ⎠
T
T1
γ ⎤
nR ⎤
n
⎛T ⎞
⎛ T2 ⎞ ⎡ n
⎡ γR
−
= cP ln 1 − R
ln ⎜ 1 ⎟ = ⎢
⎥ ln T = R ⎜ ln T ⎟ ⎢ n − 1 − γ − 1 ⎥
γ
−
−
4
n
1
T2
n − 1 ⎝ T2 ⎠
⎦
⎣
⎦
2
⎝
1 ⎠ ⎣
As s2 – s1 = s3 – s2
∴
∴
or
or
or
or
T
γR
γ ⎤
⎛ T ⎞ ⎡ n
ln 2 = R ⎜ ln 2 ⎟ × ⎢
−
⎥
T1
γ −1
⎝ T1 ⎠ ⎣ n − 1 γ − 1 ⎦
γ
n
2
=
γ −1
n −1
2nγ – 2γ = nγ – n
n (γn) = 2γ
⎛ 2γ ⎞
n= ⎜
⎟ proved
⎝ γ +1⎠
Q.10.30
A certain mass of sulphur dioxide ( SO2 ) is contained in a vessel of
Solution:
0.142 m 3 capacity, at a pressure and temperature of 23.1 bar and 18°C
respectively. A valve is opened momentarily and the pressure falls
immediately to 6.9 bar. Sometimes later the temperature is again 18°C
and the pressure is observed to be 9.1 bar. Estimate the value of specific
heat ratio.
(Ans. 1.29)
Mass of SO2 before open the valve
S = 32
O → 16 × 2 = 64
m1 =
pV
2310 × 0.142
= 8.6768 kg
=
8.3143
R SO2 T
× 291
64
Mass of SO2 after closing the valve
Page 160 of 265
R SO2 = 0.12991 kJ/kg-K
Properties of Gases and Gas Mixtures
Chapter 10
m2 =
910 × 0.142
= 3.4181 kg
RSO2 × 291
If intermediate temperature is T then
p1 V1
pV
9.1 × 0.142
6.9 × 0.142
= 2 2
or
=
291
T
T1
T2
or T = 220.65 K
As valve is opened momentarily term process is adiabatic
So
or
or
∴
Q.10.31
Solution :
T2
⎛p ⎞
= ⎜ 2⎟
T1
⎝ p1 ⎠
γ −1
γ
220.65
⎛ 6.9 ⎞
or
= ⎜
⎟
291
⎝ 23.1 ⎠
γ −1
γ
⎛ 220.65 ⎞
ln ⎜
⎟
1⎞
⎛
⎝ 299 ⎠ = 0.22903
1
−
=
⎜
γ ⎟⎠
⎛ 6.9 ⎞
⎝
ln ⎜
⎟
⎝ 23.1 ⎠
1
= 1 – 0.22903 = 0.77097
γ
γ = 1.297
A gaseous mixture contains 21% by volume of nitrogen, 50% by volume of
hydrogen, and 29% by volume of carbon-dioxide. Calculate the molecular
weight of the mixture, the characteristic gas constant R for the mixture
and the value of the reversible adiabatic index γ . (At 10ºC, the c p values
of nitrogen, hydrogen, and carbon dioxide are l.039, 14.235, and 0.828
kJ/kg K respectively.)
A cylinder contains 0.085 m 3 of the mixture at 1 bar and 10°C. The gas
undergoes a reversible non-flow process during which its volume is
reduced to one-fifth of its original value. If the law of compression is
pv1.2 = constant, determine the work and heat transfer in magnitude and
sense and the change in entropy.
(Ans. 19.64 kg/kg mol, 0.423 kJ/kg K, 1.365,
–16 kJ, – 7.24 kJ, – 0.31 kJ/kg K)
Volume ratio = 21: 50: 29
∴
Mass ratio = 21 × 28: 50 × 2: 29 × 44
Let
mN2 = 21 × 28 kg,
m H2 = 50 × 2 kg,
mN2 = 29 × 44 kg
= 588 kg
∴
Rmix =
= 100 kg
= 1276 kg
⎡
⎢RN2 =
⎢
⎢
⎢ R H2 =
⎣
m N2 R N2 + m H2 R H2 + mCO2 R CO2
m N2 + m H2 + mCO2
⎤
R
⎥
28
⎥
R
R⎥
, RCO2 =
⎥
2
44 ⎦
21 × R + 50 × R + 29 R
21 × 28 + 50 × 2 + 29 × 44
= 0.42334 kJ/kg – K
=
c p Mix =
m N2 C pN + m H2 C pH + mCO2 C pCO
2
2
2
m N2 + m H2 + mCO2
Page 161 of 265
[mN2 + m H2 + mCO2 = 1964]
Properties of Gases and Gas Mixtures
Chapter 10
=
21 × 28 × 1.039 + 100 × 14.235 + 0.828 × 1276
= 1.5738 kJ/kg – K
588 + 100 + 1276
R
= 0.74206
28
R
c VH2 = 14.235 −
= 10.078
2
R
c VCO2 = 0.828 −
= 0.63904
44
c VN2 = 1.039 −
∴
∴
588 × 0.74206 + 100 × 10.078 + 1276 × 0.63904
= 1.1505 kJ/kg – K
588 + 100 + 1276
c v mix = c P mix – Rmix = 1.5738 – 0.42334 = 1.1505 kJ/kg – K
cv
Mix
=
c p mix
γ mix =
= 1.368
cv mix
Given
p1 = 1 bar = 100 kPa
⇒
p2 = 690 kPa (Calculated)
v
V2 = 1 = 0.017 m3
5
T2 = 390.5 K (Calculated)
V2 = 0.085 m3
T1 = 10º C = 283 K
n
∴
∴
p2
⎛v ⎞
= ⎜ 1 ⎟ = 51.2
p1
⎝ v2 ⎠
p2 = 100 × 51.2 kPa
2
p
n −1
∴
T2
⎛p ⎞ n
= ⎜ 2⎟
T1
⎝ p1 ⎠
p V − p2 V2
W= 1 1
n −1
∴ T2 = 390.5 K
1
V
⎡
dV ⎤
⎢∵ W = ∫ pdV = C ∫ 4 ⎥
1
1 V ⎦
⎣
2
2
100 × 0.085 − 690 × 0.017
1.2 − 1
= –16.15 kJ
[i.e. work have to be given to the system)
=
– W
Q = u2 − u1 + W
m=
p1 V1
= 0.070948 kg
RT1
= mc v (T2 − T1 ) + W
= (8.7748 – 16.15) kJ
= –7.3752 kJ
[i.e. Heat flow through system]
Page 162 of 265
– Q
Properties of Gases and Gas Mixtures
Chapter 10
Charge of entropy
⎛T ⎞
⎛p ⎞
ΔS = S2 – S1 = mc P ln ⎜ 2 ⎟ − m R ln ⎜ 2 ⎟
⎝ T1 ⎠
⎝ p1 ⎠
⎡
⎛ 390.5 ⎞
⎛ 690 ⎞ ⎤
= m ⎢1.5738 ln ⎜
⎟ − 0.42334 × ln ⎜
⎟ ⎥ kJ/K
⎝ 283 ⎠
⎝ 100 ⎠ ⎦
⎣
= –0.022062 kJ/K = –22.062 J/K
Q.10.32
Solution:
Two moles of an ideal gas at temperature T and pressure p are contained
in a compartment. In an adjacent compartment is one mole of an ideal
gas at temperature 2Tand pressure p. The gases mix adiabatically but do
not react chemically when a partition separating: the compartments are
withdrawn. Show that the entropy increase due to the mixing process is
given by
⎛ 27
32 ⎞
γ
+
R ⎜ ln
ln
⎟
4 γ − 1 27 ⎠
⎝
Provided that the gases are different and that the ratio of specific heat γ
is the same for both gases and remains constant.
What would the entropy change be if the mixing gases were of the same
Species?
nRT
2 RT
nR 2 T
2 RT
VA =
=
VB =
=
p
p
p
p
A
B
2 mole
T
p
1 mole
2T
p
After mixing if final temperature is Tf then
2 × T + 1× 2T
4
Tf =
= T
3
2 +1
∴ Final pressure = p
4
Temperature = T
3
nRTf
=
pf =
Vf
and
After mixing Partial Pressure of A = pfA =
Partial pressure of B = pf B =
∴
9
T×p
3
4 RT
3×R×
Volume = VA + VB =
2
p
3
1
p
3
T
p ⎤
⎡
(ΔS)A = n A ⎢c pA ln f − R ln fA ⎥
TA
pA ⎦
⎣
4
2⎤
⎡ γ
= 2R ⎢
ln − ln ⎥
3
3⎦
⎣γ −1
Page 163 of 265
c PA =
γ
R
γ −1
4 RT
p
Properties of Gases and Gas Mixtures
Chapter 10
T
p ⎤
⎡
(ΔS)B = n B ⎢c PB ln f − R ln fB ⎥
TB
pB ⎦
⎣
2
1⎤
⎡ γ
= R⎢
ln − ln ⎥
3
3⎦
⎣γ −1
∴
(ΔS) univ = (ΔS)A + (ΔS)B
⎡⎛ 9
γ ⎛ 16
2 ⎞⎤
⎞
= R ⎢⎜ ln + ln 3 ⎟ +
+ ln ⎟ ⎥
⎜ ln
9
3 ⎠⎦
⎠ γ −1 ⎝
⎣⎝ 4
γ
32 ⎤
⎡ 27
= R ⎢ ln
+
ln
Proved.
γ
−
4
1
27 ⎥⎦
⎣
Q.10.33
n1 moles of an ideal gas at pressure p1 and temperature T are in one
compartment of an insulated container. In an adjoining compartment,
separated by a partition, are n2 moles of an ideal gas at pressure p2 and
temperature T. When the partition is removed, calculate (a) the final
pressure of the mixture, (b) the entropy change when the gases are
identical, and (c) the entropy change when .the gases are different.
Prove that the entropy change in (c) is the same as that produced by two
independent free expansions.
Solution:
Try please.
Q.10.34
Assume that 20 kg of steam are required at a pressure of 600 bar and a
temperature of 750°C in order to conduct a particular experiment. A 140litre heavy duty tank is available for storage.
Predict if this is an adequate storage capacity using:
(a) The ideal gas theory,
(b) The compressibility factor chart,
(c) The van der Waals equation with a = 5.454 (litre) 2 atm/ (g mol) 2 , b =
0.03042 litres/gmol for steam,
(d) The Mollier chart
(e) The steam tables.
Estimate the error in each.
Solution:
Try please.
Q.10.35
Estimate the pressure of 5 kg of CO2 gas which occupies a volume of 0.70
m 3 at 75°C, using the Beattie-Bridgeman equation of state.
Compare this result with the value obtained using the generalized
compressibility chart. Which is more accurate and why?
For CO2 with units of atm, litres/g mol and K, A o = 5.0065, a = 0.07132, Bo =
0.10476, b = 0.07235, C * 10-4 = 66.0.
Solution:
Try please.
Q.10.36
Measurements of pressure and temperature at various stages in an
adiabatic air turbine show that the states of air lie on the line pv1.25 =
Page 164 of 265
Properties of Gases and Gas Mixtures
Chapter 10
constant. If kinetic and gravitational potential energy is neglected,
prove that the shaft work per kg as a function of pressure is given by the
following relation
⎡ ⎛ p ⎞1/5 ⎤
W = 3.5p1 v1 ⎢1 – ⎜ 2 ⎟ ⎥
⎢⎣ ⎝ p1 ⎠ ⎥⎦
Take γ for air as 1.4.
Solution:
Using S.F.E.E.
⎡ V2
⎤
Q − W + Δ ⎢ 2 + g Z ⎥ = h2 – h1
⎣ 2
⎦
or
Q – W = mc p (T2 − T1 )
=
γ
mR(T2 − T1 )
γ −1
∴ p1 v1 = mRT1
p2 v 2 = mRT2
γ p1 v1 ⎡ p2 v 2
⎤
− 1⎥
⎢
γ − 1 ⎣ p1 v1
⎦
n −1
⎡
⎤
γ
⎛p ⎞ n
=
− 1⎥
p1 v1 ⎢⎜ 2 ⎟
⎢⎣⎝ p1 ⎠
⎥⎦
γ −1
=
∴ Q → 0 and as
Here adiabatic process
So
n −1
⎡
⎤
n
p
γ
⎛
⎞
2
⎢
⎥
× p1 v1 1 − ⎜ ⎟
W=
⎢⎣ ⎝ p1 ⎠
⎥⎦
γ −1
γ = 1.4 and n = 1.25
1
⎡
⎤
5
p
⎛
⎞
2
W = 3.5 p1 v1 ⎢1 − ⎜ ⎟ ⎥ proved
⎢⎣ ⎝ p1 ⎠ ⎥⎦
Q.10.37
Solution:
A mass of an ideal gas exists initially at a pressure of 200 kPa,
temperature 300 K, and specific volume 0.5 m 3 /kg. The value of r is 1.4. (a)
Determine the specific heats of the gas. (b) What is the change in entropy
when the gas is expanded to pressure 100 kPa according to the law pv1.3 =
const? (c) What will be the entropy change if the path is pv1.5 = const. (by
the application of a cooling jacket during the process)? (d) What is the
inference you can draw from this example?
(Ans. (a) 1.166,0.833 kJ/kg K, (b) 0.044 kJ/kg K (c) - 0.039 kJ/kg K
(d) Entropy increases when n < γ and decreases when n > γ )
p1 = 200 kPa
Given
T1 = 300 K
v1 = 0.5 m3/kg
γ = 1.4
(a) Gas constant( R) =
p1 v1
200 × 0.5
=
= 0.33333 kJ/kg – K
300
T1
Page 165 of 265
Properties of Gases and Gas Mixtures
Chapter 10
∴ Super heat at constant Pressure
γ
1.4
R =
× 0.33333 = 1.1667 kJ/kg – K
cp =
1.4 − 1
γ −1
c V = c p – R = 0.83333 kJ/kg – K
(b) Given p2 = 100 kPa
1
∴ s2
⎛ p ⎞1.3
v 2 = v1 × ⎜ 1 ⎟ = 0.85218 m3/kg
⎝ p2 ⎠
V
p
– s1 = c p ln 2 + cV ln 2
V1
p1
⎛ 0.85218 ⎞
⎛ 100 ⎞
= 1.1667 × ln ⎜
⎟ + 0.83333 × ln ⎜
⎟ kJ/kg − K
⎝ 0.5 ⎠
⎝ 200 ⎠
= 0.044453 kJ/kg – K = 44.453 J/kg – K
p
2
V
(c) If path is pv1.5 = C. Then
1
⎛ p ⎞1.5
v2 = v1 × ⎜ 1 ⎟ = 0.7937 m3/kg
⎝ p2 ⎠
⎛ 0.7937 ⎞
⎛ 150 ⎞
s2 – s1 = 1.1667 × ln ⎜
∴
⎟ + 0.83333 ln ⎜
⎟ kJ/kg − K
⎝ 0.5 ⎠
⎝ 200 ⎠
= –0.03849 kJ/kg – K
(d) n > γ is possible if cooling arrangement is used and ΔS will be –ve
Q.10.38
Solution:
(a) A closed system of 2 kg of air initially at pressure 5 atm and
temperature 227°C, expands reversibly to pressure 2 atm following
the law pv1.25 = const. Assuming air as an ideal gas, determine the
work done and the heat transferred.
(Ans. 193 kJ, 72 kJ)
(b) If the system does the same expansion in a steady flow process, what
is the work done by the system?
(Ans. 241 kJ)
Given
m = 2 kg
p1 = 5 atm = 506.625 kPa
1
T1 = 277º C = 550 K
p2 = 2 atm = 202.65 kPa
p
n −1
⎛p ⎞ n
T2 = T1 ⎜ 2 ⎟
= 457.9 K
⎝ p1 ⎠
p V − p2 V2
mR(T1 − T2 )
=
W1 – 2 = 1 1
n −1
n −1
2 × 0.287(550 − 457.9)
=
= 211.46 kJ
1.25 − 1
Reversible polytropic process
Heat transfer
Q1 – 2 = u2 − u1 + W1 – 2
= mc v (T2 − T1 ) + W1 – 2
Page 166 of 265
2
V
Properties of Gases and Gas Mixtures
Chapter 10
= 2 × 0.718 (457.9 – 550) + W = –132.26 + 211.46 = 79.204 kJ
(b)
Q.10.39
Solution:
For steady flow reversible polytropic process
W = h1 − h 2
n
mR
=
[p1 V1 − p2 V2 ] =
[T1 − T2 ] = 264.33 kJ
n −1
n −1
Air contained in a cylinder fitted with a piston is compressed reversibly
according to the law pv1.25 = const. The mass of air in the cylinder is 0.1
kg. The initial pressure is 100 kPa and the initial temperature 20°C. The
final volume is 1/ 8 of the initial volume. Determine the work and the
heat transfer.
(Ans. – 22.9 kJ, –8.6 kJ)
It is a reversible polytropic process
p2 = 1345.4 kPa
m = 0.1 kg
p1 = 100 kPa
T2 = 492.77 K
mRT1
∴
V1 =
V2 = 0.010511 m3
P1
= 0.084091 m3
1.25
⎛V ⎞
2
∴
p2 = p1 ⎜ 1 ⎟ = 100 × 81.25
V
⎝ 2⎠
p
n −1
⎛ p2 ⎞ n
T2 = T1 ⎜ ⎟
1
⎝ p1 ⎠
p1 V1 − p2 V2
n −1
100 × 0.084091 − 1345.4 × 0.010511
=
1.25 − 1
= –22.93 kJ
Q1 – 2 = u2 − u1 + W1 – 2
∴ W1 – 2 =
V
= mc v (T2 − T1 ) + W1 – 2
= 0.2 × 0.718 × (492.77 – 293) – 22.93
= –8.5865 kJ
Q.10.40
Solution:
Air is contained in a cylinder fitted with a frictionless piston. Initially
the cylinder contains 0.5 m 3 of air at 1.5 bar, 20°C. The air is
Then compressed reversibly according to the law pv n = constant until
the final pressure is 6 bar, at which point the temperature is 120°C.
Determine: (a) the polytropic index n, (b) the final volume of air, (c) the
work done on the air and the heat transfer, and (d) the net change in
entropy.
(Ans. (a) 1.2685, (b) 0.1676 m3 (c) –95.3 kJ, –31.5 kJ, (d) 0.0153 kJ/K)
Given
p1 = 1.5 bar = 150 kPa
T1 = 20ºC = 293 K
V1 = 0.5 m3
Page 167 of 265
Properties of Gases and Gas Mixtures
Chapter 10
p1 V1
= 0.89189 kg
RT1
p2 = 6 bar = 600 kPa
T2 = 120º C = 393 K
2
∴m=
T
⎛p ⎞
∴ 2 = ⎜ 2⎟
T1
⎝ p1 ⎠
p
n −1
n
1
V
⎛T ⎞
ln ⎜ 2 ⎟
1⎞
⎛
⎝ T1 ⎠ = 0.2118
or ⎜1 − ⎟ =
⎝
n⎠
⎛P ⎞
ln ⎜ 2 ⎟
⎝ P1 ⎠
∴ n = 1.2687
(a) The polytropic index, n = 1.2687
mRT2
0.189 × 0.287 × 393 3
m = 0.16766 m3
(b) Final volume of air (V1) =
=
600
p2
2
(c)
=
W1 – 2
∫ pdV
1
=
=
p1 V1 − p2 V2
n −1
150 × 0.5 − 600 × 0.16766
kJ = –95.263 kJ
1.2687 − 1
Q1 – 2 = u2 − u1 + W1 – 2
= mc v (T2 − T1 ) + W1 – 2
= 0.89189 × 0.718(393 – 293) + W1 – 2
= –31.225 kJ
(d)
Q.10.41
Solution:
p
V ⎤
⎡
Δs = s2 − s1 = m ⎢c v ln 2 + c P ln 2 ⎥
p1
V1 ⎦
⎣
⎡
⎛ 600 ⎞
⎛ 0.16766 ⎞ ⎤
= 0.89189 ⎢0.718 ln ⎜
⎟ + 1.005 × ln ⎜
⎟ ⎥ = –0.091663 kJ/K
⎝ 150 ⎠
⎝ 0.5 ⎠ ⎦
⎣
The specific heat at constant pressure for air is given by
c p = 0.9169 + 2.577 + 10-4 T - 3.974 * 10-8 T2 kJ/kg K
Determine the change in internal energy and that in entropy of air when
it undergoes a change of state from 1 atm and 298 K to a temperature of
2000 K at the same pressure.
( Ans. 1470.4 kJ/kg, 2.1065 kJ/kg K )
p1 = p2 = 1 atm = 101.325 kPa
T1 = 298 K; T2 = 2000 K
c p = 0.9169 + 2.577 × 10–4 T – 3.974
× 10–3 T2 kJ/kg – K
Δu = u2 − u1 =
∫m c
v
dT
Page 168 of 265
Properties of Gases and Gas Mixtures
Chapter 10
=
=
∫ m(c − R) dT
∫ mc dT − mR ∫ dT
P
P
2000
= 1×
∫
2
T
(0.9169 + 2.577 × 10 −4 T − 3.974
1
298
2000
× 10 −8 T 2 ) dT − 1 × 0.287
∫
dT kJ / kg
S
298
= (1560.6 + 503.96 – 105.62 – 488.47) kJ/kg
= 1470.5 kJ/kg
∴
Tds = dh – vdp
or
Tds = mc PdT − v dp
2
2000
dT
T
1
298
2000
+ 2.577 × 10 −4 (2000 − 298)
∴ s2 – s1 = 0.9169 × ln
298
∴
∫ dS = m
∫
cP
⎛ 20002 − 2982 ⎞
− 3.974 × 10 −8 × ⎜
⎟
2
⎝
⎠
= 2.1065 kJ/kg – K
Q.10.42
Solution:
A closed system allows nitrogen to expand reversibly from a volume of
0.25 m 3 to 0.75 m 3 along the path pv1.32 = const. The original pressure of
the gas is 250 kPa and its initial temperature is 100°C.
(a) Draw the p-v and T-s diagrams.
(b) What are the final temperature and the final pressure of the gas?
(c) How much work is done and how much heat is transferred?
(d) What is the Entropy change of nitrogen?
(Ans. (b) 262.44 K, 58.63 kPa,
(c) 57.89 kJ, 11.4 kJ, (d) 0.0362 kJ/K)
Given
p1 = 250 kPa
V1 = 0.25 m3
T1 = 100ºC = 373 K
p1
1
2
T
p
2
V
∴
m=
p2
1
p1 v1
= 0.563 kg = 0.5643 kg
RT1
Page 169 of 265
S
Properties of Gases and Gas Mixtures
Chapter 10
8.3143
= 0.29694 kJ/kg
28
n
⎛ v1 ⎞
p2 = p1 × ⎜ ⎟ = 58.633 kPa
⎝ v2 ⎠
RN2 =
V2 = 0.75 m3
n −1
⎛p ⎞ n
= 262.4 K
T2 = T1 × ⎜ 2 ⎟
⎝ p1 ⎠
p V − p2 V2
250 × 0.25 − 58.633 × 0.75
=
W= 1 1
= 57.891 kJ
n −1
(1.32 − 1)
Q = u2 − u1 + W = mc v (T2 − T1 ) + W
= 0.5643 × 0.7423 (262.4 – 373) + W
R
cv =
= 0.7423
γ −1
γ= 1+
= 11.56 kJ
cp =
∴
Q.10.43
Solution:
γ
1.4
R =
× 0.29694 = 1.04 kJ/kg – K
1.4 − 1
γ −1
2
= 1.4
5
V
p ⎤
⎡
Δs = s2 – s1 = m ⎢cP ln 2 + cV ln 2 ⎥
V1
p1 ⎦
⎣
⎡
⎛ 0.75 ⎞
⎛ 58.633 ⎞ ⎤
= 0.5643 ⎢1.04 × ln ⎜
⎟ + 0.7423 × ln ⎜
⎟ ⎥ kJ/K
⎝ 0.25 ⎠
⎝ 250 ⎠ ⎦
⎣
= 0.0373 kJ/kg – K
Methane has a specific heat at constant pressure given by c p = 17.66 +
0.06188 T kJ/kg mol K when 1 kg of methane is heated at constant
volume from 27 to 500°C. If the initial pressure of the gas is 1 atm,
calculate the final pressure, the heat transfer, the work done and the
change in entropy.
(Ans. 2.577 atm, 1258.5 kJ/kg, 2.3838 kJ/kg K)
R
Given p1 = 1 atm = 101.325 kPa
R =
16
T1 = 27ºC = 300 K
= 0.51964 kJ/kg – K
p2 = 261 kPa
m = 1 kg
mRT1
∴
V1 =
V2 = 1.5385
p1
T2 = 500ºC = 773 K
= 1.5385 m3 = V2
V=C
mRT2
(i) Find pressure (p2 ) =
V2
T
= 261 kPa ≈ 2.577 atm
(ii) Heat transfer Q =
∫ mc
v
dT
= m ∫ [c P − R]dT
S
Page 170 of 265
Properties of Gases and Gas Mixtures
Chapter 10
773
= 1×
∫ (1.1038 + 3.8675 × 10
−3
− 0.51964) dT
300
= 0.58411(773 − 300) + 3.8675 × 10 −3
(7732 − 3002 )
2
17.66 0.06188
+
T kJ/kg − K
16
16
= 1.1038 + 3.8675 × 10–3 T = 1257.7 kJ/kg
cP =
2
(iii)
Work done =
∫ pdV = 0
1
∴
Tds = du = mc v dT
dT
dT
= m(c p − R)
T
T
2
773
⎛ 1 × 0.58411 + 3.8675 × 10 −3 T ⎞
dS
=
∴
⎟ dT
∫1
∫ ⎜⎝
T
⎠
300
773
s2 – s1 = 0.58411 ln
+ 3.8675 × 10 −3 (773 − 300) = 2.3822 kJ/kg – K
300
ds = mc v
Q.10.44
Air is compressed reversibly according to the law pv1.25 = const. from an
Solution:
initial pressures of 1 bar and volume of 0.9 m 3 to a final volume of 0.6
m 3 .Determine the final pressure and the change of entropy per kg of
air.
(Ans. 1.66 bar, –0.0436 kJ/kg K)
p1 = 1 bar
V1 = 0.9 m3
V2 = 0.6 m3
1.25
⎛ V1 ⎞
∴
= 1.66 bar
p2 = p1 ⎜
⎟
⎝ V2 ⎠
2
2
T
p
1
1
V
S
V
p ⎞
⎛
Δs = s2 − s1 = ⎜ c p ln 2 + cv ln 2 ⎟
V1
p1 ⎠
⎝
⎛ 0.6 ⎞
⎛ 1.66 ⎞
= 1.005 × ln ⎜
⎟ kJ/kg − K
⎟ + 0.718 × ln ⎜
⎝ 1 ⎠
⎝ 0.9 ⎠
= –0.043587 kJ/kg – K
Q.10.45
In a heat engine cycle, air is isothermally compressed. Heat is then
added at constant pressure,
after
Page
171which
of 265 the air expands isentropically to
Properties of Gases and Gas Mixtures
Chapter 10
its original state. Draw the cycle on p-v and T'-s coordinates. Show that
the cycle efficiency can be expressed in the following form
η =1−
Solution:
( γ − 1) lnr
γ ⎡⎣ r γ −1/ γ − 1⎤⎦
Where r is the pressure ratio, p2 /p1 . Determine the pressure ratio and
the cycle efficiency if the initial temperature is 27°C and the maximum
temperature is 327°C.
( Ans. 13.4, 32.4%)
Heat addition (Q1) = Q2 – 3 = mc p (T3 − T2 )
p2
p
2
3
p1
3
W
T
2
1
1
S
V
⎛p ⎞
Heat rejection (Q2) = mRT1 ln ⎜ 2 ⎟
⎝ p1 ⎠
⎛p ⎞
RT1 ln ⎜ 2 ⎟
Q
⎝ p1 ⎠
∴
η= 1− 2 = 1−
C p (T3 − T2 )
Q1
⎛p ⎞
ln ⎜ 2 ⎟
γ −1
⎝ p1 ⎠
= 1−
γ ⎛ T3
⎞
⎜ T − 1⎟
⎝ 1
⎠
Here,
p2
=r
p1
∴
T3
⎛p ⎞
= ⎜ 3⎟
T1
⎝ p1 ⎠
cp =
= 1−
⇒
And
γ −1
γ
∴
(r
Proved
− 1)
If initial temperature (T1) = 27ºC = 300 K = T2
T3 = 327ºC = 600 K
γ
∴
ln r
γ −1
γ
1.4
⎛ T ⎞γ − 1
⎛ 600 ⎞1.4 − 1
r= ⎜ 3⎟
= ⎜
= 11.314
⎟
⎝ 300 ⎠
⎝ T1 ⎠
(1.4 − 1)
ln (11.314)
η= 1−
×
1.4 − 1
(1.4)
[ (11.314) 1.4 − 1
Page 172 of 265
]
= 0.30686
γR
γ −1
γ −1
γ
= r
γ −1
γ
Properties of Gases and Gas Mixtures
Chapter 10
Q.10.46
Solution:
What is the minimum amount of work required to separate 1 mole of
air at 27°C and 1 atm pressure (assumed composed of 1/5 O2 and 4/5 N2 )
into oxygen and nitrogen each at 27°C and 1 atm pressure?
( Ans. 1250 J)
Total air is 1 mole
1
So
O2 = mole = 0.0064 kg
5
4
N2 = mole = 0.0224 kg
5
Mixture, pressure = 1 atm, temperature = 300 K
1
Partial pressure of O2 = atm
5
4
Partial pressure of N2 = atm
5
Minimum work required is isothermal work
pf O
⎛ pf N 2 ⎞
= mO2 R O2 T1O ln 2 + mN2 R N2 T12 ln ⎜
2
⎜ p1 N ⎟⎟
p1O
2 ⎠
⎝
2
8.3143
8.3143
⎛5⎞
× 300 ln (5) + 0.0224 ×
× 300 ln ⎜ ⎟
= 0.0064 ×
32
28
⎝4⎠
= 1.248 kJ = 1248 J
A closed adiabatic cylinder of volume 1 m 3 is divided by a partition into
two compartments 1 and 2. Compartment 1 has a volume of 0.6 m 3 and
contains methane at 0.4 MPa, 40°C, while compartment 2 has a volume of
0.4 m 3 and contains propane at 0.4 MPa, 40°C. The partition is removed
and the gases are allowed to mix.
(a) When the equilibrium state is reached, find the entropy change of
the universe.
(b) What are the molecular weight and the specific heat ratio of the
mixture?
The mixture is now compressed reversibly and adiabatically to 1.2
MPa. Compute
(c) the final temperature of the mixture,
(d) The work required per unit mass, and
(e) The specific entropy change for each gas. Take c p of methane and
Q.10.47
Solution:
propane as 35.72 and 74.56 kJ/kg mol K respectively.
(Ans. (a) 0.8609 kJ/K, (b) 27.2,1.193 (c) 100.9°C,
(d) 396 kJ, (e) 0.255 kJ/kg K)
After mixing
pf = 400 kPa
Tf = 313 K
1
But partial pressure of (p1f )
CH4 =
∴
0.6
× 400 = 240 kPa
1
p2f = 0.4 × 400 = 160 kPa
Page 173 of 265
V1 = 0.6 m3
p1 = 400 kPa
T1 = 313 K
CH4
2
V2 = 0.4 m3
p2 = 400 kPa
T2 = 313 K
C3 H8
Properties of Gases and Gas Mixtures
Chapter 10
⎡
T
p ⎤
( ΔS)CH4 = mCH4 ⎢c PCH ln 2 − R ln 2 ⎥
4
T1
p1 ⎦
⎣
⎛p ⎞
= mCH4 RCH4 ln ⎜ i ⎟
⎝ pf ⎠
(a)
=
⎛p ⎞
p1 V1
× RCH4 ln ⎜ i ⎟
RCH4 T1
⎝ pf ⎠
⎛p
p1 V1
× RCH4 ln ⎜ i
⎜ pf
T1
⎝ 1
p2 V2
pi
=
× ln
T2
pf2
=
( ΔS)C3H8
⎞
⎟
⎟
⎠
(ΔS) Univ = ( ΔS)CH4 + ( ΔS)C3H8
400 × 0.6 ⎛ 400 ⎞ 400 × 0.4 ⎛ 400 ⎞
ln ⎜
ln ⎜
⎟+
⎟ kJ /K
313
313
⎝ 240 ⎠
⎝ 160 ⎠
= 0.86 kJ/K
=
(b)
Molecular weight
xM = x1M1 + x2M2
x
x
∴
M = 1 M1 + 2 × M2 = 0.6 × 16 + 0.4 × 44 = 27.2
x
x
n1c p1 + n 2 c p2
0.6 × 35.72 + 0.4 × 74.56
=
= 51.256 kJ/kg
c p mix =
1
n1 + n 2
Rmix = R = 8.3143
∴
∴
Q.10.48
c v mix = c P mix – R = 42.9417
γ mix
c P mix
=
c v mix
=
51.256
= 1.1936
42.9417
An ideal gas cycle consists of the following reversible processes: (i)
isentropic compression, (ii) constant volume heat addition, (iii)
isentropic expansion, and (iv) constant pressure heat rejection. Show
that the efficiency of this cycle is given by
(
)
1/ γ
1 ⎡γ a −1 ⎤
⎢
⎥
rkγ −1 ⎢ a − 1 ⎥
⎣
⎦
Where rk is the compression ratio and a is the ratio of pressures after
and before heat addition.
An engine operating on the above cycle with a compression ratio of 6
starts the compression with air at 1 bar, 300 K. If the ratio of pressures
after and before heat addition is 2.5, calculate the efficiency and the
m.e.p. of the cycle. Take
γ = 1.4 and c v = 0.718 kJ/kg K.
η =1−
( Ans. 0.579, 2.5322 bar)
Solution:
Q2 – 3 = u3 − u2 + pdV = mc v (T3 − T2 )
Page 174 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Q1 – 4 = mc p (T4 − T1 )
∴
η= 1−
m c p (T4 − T1 )
m c v (T3 − T2 )
⎛ T − T1 ⎞
= 1 − γ⎜ 4
⎟
⎝ T3 − T2 ⎠
γ −1
∴
⎛v ⎞
T2
= ⎜ 1⎟
= rkγ − 1
T1
v
⎝ 2⎠
T2 = T1 × rkγ − 1
γ −1
⎛p ⎞ γ
T3
= ⎜ 3⎟
T4
⎝ p4 ⎠
p3
∴
=9
p2
p v
p2 v 2
= 3 3
T2
T3
T
∴ 3 = (a × r)
T4
⎛p ⎞
= ⎜ 3⎟
⎝ p1 ⎠
V=C
3
Q1
W
γ −1
γ
γ
γ −1
γ
⎛ ⎛ V ⎞γ 1 ⎞
= ⎜⎜ 2 ⎟ × ⎟
⎜ ⎝ V1 ⎠ a ⎟
⎝
⎠
= (rk− γ a −1 )
T4 = rk1 − γ . a
∴
1−γ
k
1
γ
= r
.a
γ −1
γ
⎛p
p ⎞
= ⎜ 1 × 2⎟
⎝ p2 p3 ⎠
η= 1−
S
T
Q1
2
. a × T1 . rkγ − 1
1
Q2
S
1
γ
γ(a T1 − T1 )
(aT1rkγ − 1 − T1rkγ − 1 )
1
[ γ(a γ − 1)]
= 1 − γ −1
Proved.
rk (a − 1)
Given p1 = 1 bar = 100 kPa
T1 = 300K,
a = 2.5,
Q2
3
= a . T1
∴
4
γ −1
γ
. T3
1
−1
γ
1
γ −1
γ
r−1
r
1−γ
γ
W
p
T
∴ 3 = 3 = a,
p2
T2
⎛p ⎞
= ⎜ 1⎟
⎝ p3 ⎠
p=C
2
T
v ⎞
p2
= 1 ⎟ = aγ
p1
v2 ⎠
∴ T3 = aT2 = aT1rkγ − 1
⎛p ⎞
T4
= ⎜ 4⎟
T3
⎝ p3 ⎠
γ −1
γ
rk = 6,
γ = 1.4
Page 175 of 265
4
Properties of Gases and Gas Mixtures
Chapter 10
1
1.4(2.51.4 − 1)
∴ η = 1 − 1.4 − 1
= 0.57876
6
(2.5 − 1)
Q1 = mc v (T3 − T2 )
= m c v (aT1rkγ − 1 − T1rkγ − 1 )
= m c v T1rkγ − 1 (a − 1)
= m × 0.718 × 300 × 60.4 × (2.5 – 1)
= 661.6 m kJ
∴ W = η Q1 = 382.9 m kJ
1
For V4 = ; T4 = 2.51.4 × 300 = 577.25 K
p4 = p1 = 100 kPa
mRT4
m × 0.287 × 577.25 3
m
V4 =
=
p4
100
= 1.6567 m m3
m R T1
m × 0.287 × 300
=
V1 =
p1
100
= 0.861m m3
∴
V4 – V1 = 0.7957 m3
Let m.e.p. is pm then
pm ( V4 – V1 ) = W
382.9 × m
kPa
pm =
0.7957 m
= 481.21 kPa = 4.8121 bar
Q10.49
Solution:
Q10.50
Solution:
Q10.51
The relation between u, p and v for many gases is of the form u = a + bpv
where a and b are constants. Show that for a reversible adiabatic
process pv y = constant, where
γ = (b + 1)/b.
Try please.
(a) Show that the slope of a reversible adiabatic process on p-v
coordinates is
dp
1 cp
1 ⎛ ∂v ⎞
=
wherek = − ⎜
⎟
dv kv c v
v ⎝ ∂p ⎠ T
(b) Hence, show that for an ideal gas, pv γ = constant, for a reversible
adiabatic process.
Try please.
A certain gas obeys the Clausius equation of state p (v – b) = RT and has
its internal energy given by u = c v T. Show that the equation for a
reversible adiabatic process is p ( v − b ) = constant, where γ = c p / c v .
γ
Solution:
Try please.
Page 176 of 265
Properties of Gases and Gas Mixtures
Chapter 10
Q10.52
Solution:
Q10.53
(a) Two curves, one representing a reversible adiabatic process
undergone by an ideal gas and the other an isothermal process by
the same gas, intersect at the same point on the p-v diagram. Show
that the ratio of the slope of the adiabatic curve to the slope of the
isothermal curve is equal to γ .
(b) Determine the ratio of work done during a reversible adiabatic
process to the work done during an isothermal process for a gas
having γ = 1.6. Both processes have a pressure ratio of 6.
Try please.
Two containers p and q with rigid walls contain two different
monatomic gases with masses m p and m q , gas constants Rp and Rq ,
and initial temperatures Tp and Tq respectively, are brought in contact
Solution:
Q10.54
Solution:
Q10.55
Solution:
with each other and allowed to exchange energy until equilibrium is
achieved. Determine:
(a) the final temperature of the two gases and
(b) the change of entropy due to this energy exchange.
Try please.
The pressure of a certain gas (photon gas) is a function of temperature
only and is related to the energy and volume by p(T) = (1/3) (U/V). A
system consisting of this gas confined by a cylinder and a piston
undergoes a Carnot cycle between two pressures P1 and P2 .
(a) Find expressions for work and heat of reversible isothermal and
adiabatic processes.
(b) Plot the Carnot cycle on p-v and T- s diagrams.
(c) Determine the efficiency of the cycle in terms of pressures.
(d) What is the functional relation between pressure and temperature?
Try please.
The gravimetric analysis of dry air is approximately: oxygen = 23%,
nitrogen = 77%. Calculate:
(a) The volumetric analysis,
(b) The gas constant,
(c) The molecular weight,
(d) the respective partial pressures,
(e) The specific volume at 1 atm, 15°C, and
(f) How much oxygen must be added to 2.3 kg air to produce . A mixture
which is 50% oxygen by volume?
(Ans. (a) 21% O2 , 79% N 2 , (b) 0.288 kJ/kg K,
By gravimetric analysis
(d) 21 kPa for O2 ' (e) 0.84 m3 /kg, (f) 1.47 kg)
O2: N2 = 23: 77
(a) ∴ By volumetric analysis O2: N2 =
23 77
:
32 28
= 0.71875: 2.75
(100)
2.75 × 100
:
(0.71875 − 2.75)
2.75
= 20.72: 79.28
Page 177 of 265
= 0.71875 ×
Properties of Gases and Gas Mixtures
Chapter 10
(b)
Let total mass = 100 kg
∴
O2 = 23 kg, N2= 77 kg
∴
R=
23 × R O2 + 77 × R N2
23 + 77
’
8.3143
8.3143
+ 77 ×
32
28
=
23 + 77
= 0.2884 kJ/kg – K
23 ×
(c)
For molecular weight (μ)
xμ = x1 μ1 + x 2 μ2
x
x
or
μ = 1 × μ1 + 2 μ2
x
x
= 2072 × 32 + 0.7928 × 28 = 28.83
(d)
Partial pressure of O2 = x O2 × p
= 0.2072 × 101.325 kPa = 20.995 kPa
Partial pressure of N2 = x N 2 × p = 0.7928 × 101.325 kPa = 80.33 kPa
(e)
RT
0.2884 × 288 3
=
m / kg = 0.81973 m3/kg
101.325
ρ
ρ = ρ1 + ρ2
pN 2
pO2
1
1
1
+
=
+
=
v
v1 v 2
R O2 × 288 R N2 × 288
Sp. volume, v =
Density
∴
0.2072 × 101.325 × 32 0.7928 × 101.325 × 28
+
8.3143 × 2.88
8.3143 × 288
3
v = 0.81974 m /kg
=
∴
(f)
In 2.3 kg of air O2 = 2.3 × 0.23 kg = 0.529 kg
∴
N2 = 2.3 × 0.77 = 1.771 kg = 63.25 mole
For same volume we need same mole O2
32
Total
O2 = 63.25 ×
kg = 2.024 kg
1000
∴ Oxygen must be added = (2.024 – 0.529) kg = 1.495 kg
Page 178 of 265
Thermodynamic Relations
Chapter 11
11. Thermodynamic Relations
Some Important Notes
Some Mathematical Theorem
Theorem 1. If a relation exists among the variables x, y and z, then z may be expressed as a
function of x and y, or
⎛ ∂z ⎞
⎛ ∂z ⎞
dz = ⎜ ⎟ dx + ⎜ ⎟ dy
⎝ ∂x ⎠ y
⎝ ∂y ⎠ x
then dz = M dx + N dy.
Where z, M and N are functions of x and y. Differentiating M partially with respect to y, and N
with respect to x.
∂2 z
⎛ ∂M ⎞
⎜ ∂y ⎟ = ∂x.∂y
⎝
⎠x
∂2 z
⎛ ∂N ⎞
⎜ ∂x ⎟ = ∂y.∂x
⎝
⎠y
⎛ ∂M ⎞
⎛ ∂N ⎞
⎜ ∂y ⎟ = ⎜ ∂x ⎟
⎠y
⎝
⎠x ⎝
This is the condition of exact (or perfect) differential.
Theorem 2. If a quantity f is a function of x, y and z, and a relation exists among x, y and z,
then f is a function of any two of x, y and z. Similarly any one of x, y and z may be regarded to
be a function of f and any one of x, y and z. Thus, if
x = x (f, y)
⎛ ∂x ⎞
⎛ ∂x ⎞
dx = ⎜ ⎟ df + ⎜ ⎟ dy
⎝ ∂f ⎠ y
⎝ ∂y ⎠ f
Similarly, if
y = y (f, z)
⎛ ∂y ⎞
⎛ ∂y ⎞
dy = ⎜ ⎟ df + ⎜ ⎟ dz
⎝ ∂z ⎠ f
⎝ ∂f ⎠ z
Substituting the expression of dy in the preceding equation
Page 179 of 265
Thermodynamic Relations
Chapter 11
Theorem 3. Among the variables x, y, and z any one variable may be considered as a function
of the other two. Thus
x = x(y, z)
⎛ ∂x ⎞
⎛ ∂x ⎞
dx = ⎜ ⎟ dy + ⎜ ⎟ dz
⎝ ∂z ⎠ y
⎝ ∂y ⎠ z
Similarly,
⎛ ∂z ⎞
⎛ ∂z ⎞
dz = ⎜ ⎟ dx + ⎜ ⎟ dy
⎝ ∂x ⎠ y
⎝ ∂y ⎠ x
⎤
⎛ ∂x ⎞
⎛ ∂z ⎞
⎛ ∂x ⎞ ⎡⎛ ∂z ⎞
dx = ⎜ ⎟ dy + ⎜ ⎟ ⎢⎜ ⎟ dx + ⎜ ⎟ dy ⎥
⎝ ∂z ⎠ y ⎢⎣⎝ ∂x ⎠ y
⎝ ∂y ⎠ z
⎝ ∂y ⎠ x ⎥⎦
⎡ ⎛ ∂x ⎞ ⎛ ∂x ⎞ ⎛ ∂z ⎞ ⎤
⎛ ∂x ⎞ ⎛ ∂z ⎞
= ⎢ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎥ dy + ⎜ ⎟ ⎜ ⎟ dx
⎝ ∂z ⎠ y ⎝ ∂x ⎠ y
⎢⎣ ⎝ ∂y ⎠ z ⎝ ∂z ⎠ y ⎝ ∂y ⎠ x ⎥⎦
⎡ ⎛ ∂x ⎞ ⎛ ∂x ⎞ ⎛ ∂z ⎞ ⎤
= ⎢ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎥ dy + dx
⎣⎢ ⎝ ∂y ⎠ z ⎝ ∂z ⎠ y ⎝ ∂y ⎠ x ⎦⎥
⎛ ∂x ⎞ ⎛ ∂z ⎞ ⎛ ∂x ⎞
∴⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ = 0
⎝ ∂y ⎠ z ⎝ ∂y ⎠ x ⎝ ∂z ⎠ y
⎛ ∂x ⎞ ⎛ ∂z ⎞ ⎛ ∂y ⎞
⎜ ∂y ⎟ ⎜ ∂x ⎟ ⎜ ∂z ⎟ = −1
⎝ ⎠z ⎝ ⎠ y ⎝ ⎠x
Among the thermodynamic variables p, V and T. The following relation holds good
⎛ ∂p ⎞ ⎛ ∂V ⎞ ⎛ ∂T ⎞
⎜ ∂V ⎟ ⎜ ∂T ⎟ ⎜ ∂p ⎟ = −1
⎝
⎠T ⎝
⎠p ⎝
⎠v
Maxwell’s Equations
A pure substance existing in a single phase has only two independent variables. Of the eight
quantities p, V, T, S, U, H, F (Helmholtz function), and G (Gibbs function) any one may be
expressed as a function of any two others.
For a pure substance undergoing an infinitesimal reversible process
(a) dU = TdS - pdV
(b) dH = dU + pdV + VdP = TdS + Vdp
(c) dF = dU - TdS - SdT = - pdT - SdT
(d) dG = dH - TdS - SdT = Vdp - SdT
Since U, H, F and G are thermodynamic properties and exact differentials of the type
dz = M dx + N dy, then
⎛ ∂M ⎞
⎛ ∂N ⎞
⎜
⎟ =⎜
⎟
∂
y
⎝
⎠ x ⎝ ∂x ⎠ y
Applying this to the four equations
Page 180 of 265
Thermodynamic Relations
Chapter 11
⎛ ∂T ⎞
⎛ ∂p ⎞
⎜ ∂V ⎟ = − ⎜ ∂S ⎟
⎝
⎠s
⎝
⎠v
⎛ ∂T ⎞
⎛ ∂V ⎞
⎜ ∂P ⎟ = ⎜ ∂S ⎟
⎝
⎠s ⎝
⎠p
⎛ ∂p ⎞
⎛ ∂S ⎞
⎜ ∂T ⎟ = ⎜ ∂V ⎟
⎝
⎠V ⎝
⎠T
⎛ ∂S ⎞
⎛ ∂V ⎞
⎜ ∂T ⎟ = − ⎜ ∂p ⎟
⎝
⎠P
⎝
⎠T
These four equations are known as Maxwell’s equations.
Questions with Solution (IES & IAS)
(i) Derive: dS = C
v
dT ⎛ ∂p ⎞
+
dV
T ⎜⎝ ∂T ⎟⎠
[IAS - 1986]
v
Let entropy S be imagined as a function of T and V.
S = S ( T, V )
Then
⎛ ∂S ⎞
⎛ ∂S ⎞
dS = ⎜
⎟ dT + ⎜ ∂V ⎟ dV
T
∂
⎝
⎠V
⎝
⎠T
multiplying both side by T
or
Since
and
∴
⎛ ∂S ⎞
⎛ ∂S ⎞
TdS = T ⎜
dT + T ⎜
⎟
⎟ dV
⎝ ∂T ⎠ V
⎝ ∂V ⎠ T
⎛ ∂S ⎞
T⎜
⎟ = CV , heat capacity at constant volume
⎝ ∂T ⎠ V
⎛ ∂p ⎞
⎛ ∂S ⎞
⎜ ∂V ⎟ = ⎜ ∂T ⎟ by Maxwell 's equation
⎠V
⎝
⎠T ⎝
⎛ ∂p ⎞
TdS = CV dT + T ⎜
⎟ dV
⎝ ∂T ⎠ V
dividing both side by T
dS = CV
dT ⎛ ∂p ⎞
dV proved
+
T ⎜⎝ ∂T ⎟⎠ V
(ii) Derive:
⎛ ∂V ⎞
TdS = CpdT − T ⎜
⎟ dp
⎝ ∂T ⎠ p
[IES-1998]
Let entropy S be imagined as a function of T and p.
Page 181 of 265
Thermodynamic Relations
Chapter 11
S = S ( T, p)
Then
⎛ ∂S ⎞
⎛ ∂S ⎞
dS = ⎜
dT + ⎜ ⎟ dp
⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠ T
multiplying both side by T
or
⎛ ∂S ⎞
⎛ ∂S ⎞
TdS = T ⎜
dT + T ⎜ ⎟ dp
⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠ T
Since
⎛ ∂S ⎞
T⎜
⎟ = Cp , heat capacity at constant pressure
⎝ ∂T ⎠ p
and
⎛ ∂S ⎞
⎛ ∂V ⎞
⎜ ∂p ⎟ = − ⎜ ∂T ⎟ by Maxwell 's equation
⎝
⎠p
⎝ ⎠T
∴
⎛ ∂V ⎞
TdS = CpdT − T ⎜
⎟ dp
⎝ ∂T ⎠ p
proved.
(iii) Derive:
TdS = C V dT + T
k Cv dp Cp
β
dV = CpdT − TVβ dp =
dV
+
k
β
βV
[IES-2001]
We know that volume expansivity (β) =
1 ⎛ ∂V ⎞
V ⎜⎝ ∂T ⎟⎠ p
and
isothermal compressibility (k) = −
∴
From first TdS equation
1 ⎛ ∂V ⎞
V ⎜⎝ ∂p ⎟⎠ T
⎛ ∂p ⎞
TdS = CV dT + T ⎜
⎟ dV
⎝ ∂T ⎠ V
⎛ ∂V ⎞
⎜ ∂T ⎟
β
⎝
⎠p
⎛ ∂V ⎞ ⎛ ∂p ⎞
=−
= −⎜
⎟ ⋅⎜
⎟
k
⎛ ∂V ⎞
⎝ ∂T ⎠ p ⎝ ∂V ⎠ T
⎜ ∂p ⎟
⎝
⎠T
As
⎛ ∂V ⎞ ⎛ ∂T ⎞ ⎛ ∂p ⎞
⎜ ∂T ⎟ ⋅ ⎜ ∂p ⎟ ⋅ ⎜ ∂V ⎟ = − 1
⎝
⎠p ⎝
⎠T
⎠V ⎝
∴
⎛ ∂V ⎞ ⎛ ∂p ⎞
⎛ ∂p ⎞
−⎜
⋅⎜
=⎜
⎟
⎟
⎟
⎝ ∂T ⎠ p ⎝ ∂V ⎠ T ⎝ ∂T ⎠ V
or
β ⎛ ∂p ⎞
=
k ⎜⎝ ∂T ⎟⎠ V
β
⋅ dV
k
From second TdS relation
∴
TdS = C V dT + T ⋅
proved
Page 182 of 265
Thermodynamic Relations
Chapter 11
⎛ ∂V ⎞
TdS = CpdT − T ⎜
⎟ dp
⎝ ∂T ⎠ p
as
∴
∴
β=
1 ⎛ ∂V ⎞
V ⎜⎝ ∂T ⎟⎠ p
⎛ ∂V ⎞
⎜ ∂T ⎟ = Vβ
⎝
⎠p
TdS = CpdT − TVβ dp
proved
Let S is a function of p, V
∴
S = S(p, V)
∴
⎛ ∂S ⎞
⎛ ∂S ⎞
dS = ⎜ ⎟ dp + ⎜ ⎟ dV
⎝ ∂V ⎠ p
⎝ ∂p ⎠ V
Multiply both side by T
⎛ ∂S ⎞
⎛ ∂S ⎞
TdS = T ⎜ ⎟ dp + T ⎜
⎟ dV
⎝ ∂V ⎠ p
⎝ ∂p ⎠ V
or
⎛ ∂S ∂T ⎞
⎛ ∂S ∂T ⎞
⋅
TdS = T ⎜
⎟ dp + T ⎜ ∂T ⋅ ∂V ⎟ dV
∂
∂
T
p
⎝
⎠p
⎝
⎠V
or
⎛ ∂S ⎞ ⎛ ∂T ⎞
⎛ ∂S ⎞ ⎛ ∂T ⎞
TdS = T ⎜
⎟ ⋅ ⎜ ∂p ⎟ dp + T ⎜ ∂T ⎟ ⋅ ⎜ ∂V ⎟ dV
∂
T
⎝
⎠V ⎝
⎝
⎠p ⎝
⎠p
⎠V
⎛ ∂S ⎞
Cp = T ⎜
⎟
⎝ ∂T ⎠ p
∴
and
⎛ ∂S ⎞
CV = T ⎜
⎟
⎝ ∂T ⎠ V
⎛ ∂T ⎞
⎛ ∂T ⎞
TdS = Cv ⎜
dp + Cp ⎜
⎟ dV
⎟
⎝ ∂V ⎠ p
⎝ ∂p ⎠ V
β ⎛ ∂p ⎞
=
k ⎜⎝ ∂T ⎟⎠ V
From first
or
k ⎛ ∂T ⎞
=
β ⎜⎝ ∂p ⎟⎠ V
k
⎛ ∂T ⎞
dp + Cp ⎜
⎟ dV
β
⎝ ∂V ⎠ p
∴
TdS = Cv
∴
β=
∴
1
⎛ ∂T ⎞
⎜ ∂V ⎟ = βV
⎝
⎠p
∴
TdS =
1 ⎛ ∂V ⎞
V ⎜⎝ ∂T ⎟⎠ p
Cv k dp Cp
+
dV
β
βV
proved.
(iv) Prove that
2
⎛ ∂V ⎞ ⎛ ∂p ⎞
Cp − Cv = − T ⎜
⋅⎜
⎟
⎟
⎝ ∂T ⎠ p ⎝ ∂V ⎠T
We know that
Page 183 of 265
[IAS-1998]
Thermodynamic Relations
Chapter 11
⎛ ∂V ⎞
⎛ ∂p ⎞
TdS = CpdT − T ⎜
dp = C V dT + T ⎜
⎟
⎟ dV
⎝ ∂T ⎠ p
⎝ ∂T ⎠ V
(C
or
p
− Cv ) dT =
⎛ ∂V ⎞
⎛ ∂p ⎞
T⎜
dp + T ⎜
⎟
⎟ dV
⎝ ∂T ⎠ p
⎝ ∂T ⎠ V
⎛ ∂V ⎞
⎛ ∂p ⎞
T⎜
dV
⎟ dp T ⎜
T
∂
∂T ⎟⎠ V
⎝
⎠p
⎝
dT =
+
Cp − CV
Cp − C V
or
− − − (i )
sin ce T is a function of p, V
T = T ( p, V )
⎛ ∂T ⎞
⎛ ∂T ⎞
dT = ⎜
dp + ⎜
⎟ dV
⎟
⎝ ∂V ⎠ p
⎝ ∂p ⎠ V
or
− − − ( ii )
comparing ( i ) & ( ii ) we get
⎛ ∂V ⎞
T⎜
⎟
⎝ ∂T ⎠ p ⎛ ∂T ⎞
=⎜
⎟
Cp − CV
⎝ ∂p ⎠ V
both these give
⎛ ∂p ⎞
T⎜
∂T ⎠⎟ V ⎛ ∂T ⎞
and ⎝
=⎜
⎟
Cp − CV
⎝ ∂V ⎠ p
⎛ ∂V ⎞ ⎛ ∂p ⎞
Cp − CV = T ⎜
⎟ ⎜
⎟
⎝ ∂T ⎠ p ⎝ ∂T ⎠ V
Here
⎛ ∂p ⎞ ⎛ ∂T ⎞ ⎛ ∂V ⎞
⎛ ∂p ⎞
⎛ ∂V ⎞ ⎛ ∂p ⎞
⎜ ∂T ⎟ ⋅ ⎜ ∂V ⎟ ⋅ ⎜ ∂p ⎟ = − 1 or ⎜ ∂T ⎟ = − ⎜ ∂T ⎟ ⋅ ⎜ ∂V ⎟
⎝
⎠V ⎝
⎠p ⎝
⎝
⎠V
⎝
⎠p ⎝
⎠T
⎠T
∵
⎛ ∂V ⎞ ⎛ ∂p ⎞
Cp − C V = − T ⎜
⎟ ⋅⎜
⎟
⎝ ∂T ⎠ p ⎝ ∂V ⎠ T
2
proved.
...............Equation(A)
This is a very important equation in thermodynamics. It indicates the following important
facts.
⎛ ∂V ⎞
2
⎛ ∂p ⎞
(a) Since ⎜
⎟ is always positive, and ⎜ ∂V ⎟ for any substance is negative. (Cp – Cv) is always
⎝ ∂T ⎠ p
⎝
⎠T
positive. Therefore, Cp is always greater than Cv.
(b) As T → 0 K ,C p → Cv or at absolute zero, Cp = Cv.
⎛ ∂V ⎞
(c) When ⎜
⎟ = 0 (e.g for water at 4ºC, when density is maximum. Or specific volume
⎝ ∂T ⎠ p
minimum). Cp = Cv.
(d) For an ideal gas, pV = mRT
mR V
⎛ ∂V ⎞
⎜ ∂T ⎟ = P = T
⎝
⎠p
and
∴
mRT
⎛ ∂p ⎞
⎜ ∂V ⎟ = − V 2
⎝
⎠T
C p − Cv = mR
or c p − cv = R
Equation (A) may also be expressed in terms of
volume
expansively
(β) defined as
Page
184 of
265
Thermodynamic Relations
Chapter 11
β=
1 ⎛ ∂V ⎞
V ⎜⎝ ∂T ⎟⎠ p
and isothermal compressibility (kT) defined as
kT = −
1 ⎛ ∂V ⎞
⎜
⎟
V ⎝ ∂p ⎠T
⎡ 1 ⎛ ∂V ⎞ ⎤
TV ⎢ ⎜
⎟ ⎥
⎢⎣ V ⎝ ∂T ⎠ p ⎥⎦
C p − Cv =
1 ⎛ ∂V ⎞
− ⎜
V ⎝ ∂p ⎟⎠T
C p − Cv =
2
TV β 2
kT
(v) Prove that
β ⎛ ∂p ⎞
=
k ⎜⎝ ∂T ⎟⎠ V
and
⎧
⎛ ∂U ⎞ ⎫ ⎛ ∂V ⎞
Cp − CV = ⎨ p + ⎜
⎟ ⎬⎜
⎟
⎝ ∂V ⎠ T ⎭ ⎝ ∂T ⎠ p
⎩
Hence show that
β2 TV
Cp - Cv =
k
Here β =
1 ⎛ ∂V ⎞
V ⎜⎝ ∂T ⎟⎠ p
k=−
∴
[IES-2003]
1 ⎛ ∂V ⎞
V ⎜⎝ ∂p ⎟⎠ T
⎛ ∂V ⎞
⎜ ∂T ⎟
β
⎝
⎠p
⎛ ∂V ⎞ ⎛ ∂p ⎞
=−
= −⎜
⎟ ⋅⎜
⎟
k
⎛ ∂V ⎞
⎝ ∂T ⎠ p ⎝ ∂V ⎠ T
⎜ ∂p ⎟
⎝
⎠T
⎛ ∂V ⎞ ⎛ ∂T ⎞ ⎛ ∂p ⎞
⎜ ∂T ⎟ ⋅ ⎜ ∂p ⎟ ⋅ ⎜ ∂V ⎟ = − 1
⎝
⎠p ⎝
⎠T
⎠V ⎝
we know that
or
∴
⇒
⎛ ∂V ⎞ ⎛ ∂p ⎞
⎛ ∂p ⎞
−⎜
⋅⎜
=⎜
⎟
⎟
⎟
⎝ ∂T ⎠ p ⎝ ∂V ⎠ T ⎝ ∂T ⎠ V
β ⎛ ∂p ⎞
=
k ⎜⎝ ∂T ⎟⎠ V
proved.
From Tds relations
Page 185 of 265
Thermodynamic Relations
Chapter 11
⎛ ∂V ⎞
⎛ ∂p ⎞
TdS = CpdT − T ⎜
dP = CV dT + T ⎜
⎟
⎟ dV
⎝ ∂T ⎠ p
⎝ ∂T ⎠ V
∴
or
(C
p
⎛ ∂V ⎞
⎛ ∂p ⎞
dP + T ⎜
− Cv ) dT = T ⎜
⎟
⎟ dV
⎝ ∂T ⎠ p
⎝ ∂T ⎠ v
⎛ ∂V ⎞
⎛ ∂p ⎞
T⎜
T⎜
⎟
T
∂
∂T ⎟⎠ V
⎝
⎠p
dT =
dP + ⎝
dV − − − ( i )
Cp − CV
Cp − CV
S in ce T is a function of ( p, V )
T = T ( p, V )
∴
⎛ ∂T ⎞
⎛ ∂T ⎞
dT = ⎜
dp + ⎜
⎟ dV
⎟
⎝ ∂V ⎠ p
⎝ ∂p ⎠ V
− − − ( ii )
Compairing ( i ) & ( ii ) we get
⎛ ∂V ⎞
T⎜
⎟
⎝ ∂T ⎠ p ⎛ ∂T ⎞
=⎜
⎟
Cp − CV
⎝ ∂p ⎠ V
as
⎛ ∂V ⎞ ⎛ ∂p ⎞
Cp − CV = T ⎜
⎟ ⋅⎜
⎟
⎝ ∂T ⎠ p ⎝ ∂T ⎠ V
dU = dQ − pdV
∴
dU = TdS − pdV
∴
or
or
and
⎛ ∂p ⎞
T⎜
⎟
⎝ ∂T ⎠ V = ⎛ ∂T ⎞
⎜ ∂V ⎟
Cp − CV
⎝
⎠p
⎛ ∂U ⎞
⎛ ∂S ⎞
⎜ ∂V ⎟ = T ⎜ ∂V ⎟ − p
⎝
⎠T
⎝
⎠T
⎛ ∂U ⎞
⎛ ∂S ⎞
⎜ ∂V ⎟ + p = T ⎜ ∂V ⎟
⎝
⎠T
⎝
⎠T
From Maxwell 's Third relations
⎛ ∂p ⎞
⎛ ∂S ⎞
⎜ ∂T ⎟ = ⎜ ∂V ⎟
⎝
⎠V ⎝
⎠T
∴
⎧
⎛ ∂V ⎞ ⎛ ∂p ⎞
⎛ ∂U ⎞ ⎫ ⎛ ∂V ⎞
Cp − CV = T ⎜
⎟ ⋅ ⎜ ∂T ⎟ = ⎨ p + ⎜ ∂V ⎟ ⎬ ⎜ ∂T ⎟
T
∂
⎝
⎠p ⎝
⎠V ⎩
⎝
⎠T ⎭ ⎝
⎠p
(vi) Prove that
Joule – Thomson co-efficient
T2 ⎡ ∂ ⎛ V ⎞ ⎤
⎛ ∂T ⎞
μ=⎜
⎟ = C ⎢ ∂T ⎜ T ⎟ ⎥
∂
p
⎝ ⎠⎦p
⎝
⎠h
p ⎣
[IES-2002]
The numerical value of the slope of an isenthalpic on a T – p diagram at any point is called
the Joule – Kelvin coefficient.
Page 186 of 265
Thermodynamic Relations
Chapter 11
(vii) Derive Clausius – Clapeyron equation
h fg
⎛ dp ⎞
=
⎜ dT ⎟
⎝
⎠ T ( vg − v f )
⎛ ∂p ⎞
⎛ ∂S ⎞
⎜ ∂T ⎟ = ⎜ ∂V ⎟
⎝
⎠V ⎝
⎠T
h
dp
= fg2 dT
p
RT
and
[IES-2000]
Maxwells equation
When saturated liquid convert to saturated vapour at constant temperature. During the
evaporation, the pr. & T is independent of volume.
sg − s f
⎛ dp ⎞
⎜ dT ⎟ = v − v
⎝
⎠sat
g
f
∴
sg − sf = sfg =
or
h fg
T
h fg
⎛ dp ⎞
⎜ dT ⎟ =
⎝
⎠sat T ( v g − v f )
→ It is useful to estimate properties like h from other measurable properties.
→ At a change of phage we may find h fg i.e. latent heat.
At very low pressure v g ≈ v f g as v f very small
pv g = RT
∴
or
vg =
RT
p
h fg
h fg
h ⋅p
dp
=
=
= fg 2
dT T ⋅ v g T ⋅ RT
RT
p
or
dp h fg dT
=
⋅
p
R T2
or
⎛ p ⎞ h fg ⎛ 1
1 ⎞
ln ⎜ 2 ⎟ =
⎜ − ⎟
R ⎝ T1 T2 ⎠
⎝ p1 ⎠
→ Knowing vapour pressure p1 at temperature T1, we may find out p2 at temperature T2.
Page 187 of 265
Thermodynamic Relations
Chapter 11
Joule-Kelvin Effect or Joule-Thomson coefficient
The value of the specific heat cp can be determined from p–v–T data and the Joule–Thomson
coefficient. The Joule–Thomson coefficient μJ is defined as
⎛ ∂T ⎞
μJ = ⎜
⎟
⎝ ∂p ⎠h
Like other partial differential coefficients introduced in this section, the Joule–Thomson
coefficient is defined in terms of thermodynamic properties only and thus is itself a property.
The units of μJ are those of temperature divided by pressure.
A relationship between the specific heat cp and the Joule–Thomson coefficient μJ can be
established to write
⎛ ∂T ⎞ ⎛ ∂p ⎞ ⎛ ∂h ⎞
⎜ ∂p ⎟ ⎜ ∂h ⎟ ⎜ ∂T ⎟ = − 1
⎠p
⎝
⎠ h ⎝ ⎠T ⎝
The first factor in this expression is the Joule–Thomson coefficient and the third is cp. Thus
cp =
With ( ∂h / ∂p )T = 1 / ( ∂p / ∂h )T
−1
μJ ( ∂p / ∂h )T
this can be written as
cp = −
1 ⎛ ∂h ⎞
μJ ⎜⎝ ∂p ⎟⎠T
The partial derivative ( ∂h / ∂p )T , called the constant-temperature coefficient, can be
eliminated. The following expression results:
cp =
⎤
1 ⎡ ⎛ ∂v ⎞
T
v
−
⎢
⎥
μJ ⎣ ⎜⎝ ∂T ⎟⎠ p ⎦
allows the value of cp at a state to be determined using p–v–T data and the value of the Joule–
Thomson coefficient at that state. Let us consider next how the Joule–Thomson coefficient can
be found experimentally.
The numerical value of the slope of an isenthalpic on a T-p diagram at any point is called the
Joule-Kelvin coefficient and is denoted by μJ . Thus the locus of all points at which μJ is zero is
the inversion curve. The region inside the inversion curve where μJ is positive is called the
cooling region and the region outside where μJ is negative is called the heating region. So,
Page 188 of 265
Thermodynamic Relations
Chapter 11
⎛ ∂T ⎞
μJ = ⎜
⎟
⎝ ∂p ⎠h
Energy Equation
For a system undergoing an infinitesimal reversible process between two equilibrium states,
the change of internal energy is
dU = TdS - pdV
Substituting the first TdS equation
⎛ ∂p
dU = Cv dT + T ⎜
⎝ ∂T
⎞
⎟ dV − pdV
⎠V
⎡ ⎛ ∂p ⎞
⎤
= Cv dT + ⎢T ⎜
− p ⎥ dV
⎟
⎢⎣ ⎝ ∂T ⎠V
⎥⎦
if U = (T ,V )
⎛ ∂U ⎞
⎛ ∂U ⎞
dU = ⎜
⎟ dT + ⎜ ∂V ⎟ dV
⎝ ∂T ⎠V
⎝
⎠T
⎛ ∂U ⎞
⎛ ∂p ⎞
⎜ ∂V ⎟ = T ⎜ ∂T ⎟ − p
⎝
⎠T
⎝
⎠V
This is known as energy equation. Two application of the equation are given below(a) For an ideal gas, p =
nRT
V
nR p
⎛ ∂p ⎞
∴⎜
⎟ = V =T
T
∂
⎝
⎠V
p
⎛ ∂U ⎞
∴⎜
= T. − p = 0
⎟
T
⎝ ∂V ⎠T
U does not change when V changes at T = C.
⎛ ∂U ⎞ ⎛ ∂p ⎞ ⎛ ∂V ⎞
⎜
⎟ ⎜
⎟ ⎜
⎟ =1
⎝ ∂p ⎠T ⎝ ∂V ⎠T ⎝ ∂U ⎠T
⎛ ∂U ⎞ ⎛ ∂p ⎞
⎛ ∂U ⎞
=⎜
⎜
⎟ ⎜
⎟
⎟ =0
⎝ ∂p ⎠T ⎝ ∂V ⎠T ⎝ ∂V ⎠T
⎛ ∂U ⎞
⎛ ∂p ⎞
since ⎜
≠ 0, ⎜
⎟ =0
⎟
⎝ ∂V ⎠T
⎝ ∂p ⎠T
U does not change either when p changes at T = C. So the internal energy of an ideal gas is a
function of temperature only.
Another important point to note is that for an ideal gas
⎛ ∂p ⎞
pV = nRT and T ⎜
⎟ −p=0
⎝ ∂T ⎠v
Page 189 of 265
Thermodynamic Relations
Chapter 11
Therefore
dU = Cv dT
holds good for an ideal gas in any process (even when the volume changes). But for any other
substance
dU = Cv dT
is true only when the volume is constant and dV = 0
Similarly
dH = TdS + Vdp
⎛ ∂V ⎞
TdS = Cp dT − T ⎜
⎟ dp
⎝ ∂T ⎠ p
and
⎡
⎛ ∂V ⎞ ⎤
∴ dH = C p dT + ⎢V − T ⎜
⎟ ⎥ dp
⎝ ∂T ⎠ p ⎥⎦
⎢⎣
⎛ ∂H ⎞
⎛ ∂V ⎞
∴⎜
⎟ = V −T ⎜
⎟
p
∂
⎝ ∂T ⎠ p
⎝
⎠T
As shown for internal energy, it can be similarly proved from Eq. shown in above that the
enthalpy of an ideal gas is not a function of either volume or pressure.
⎡ ⎛ ∂H ⎞
⎤
⎛ ∂H ⎞
= 0⎥
⎢i.e ⎜
⎟ = 0 and ⎜
⎟
⎝ ∂V ⎠T
⎣ ⎝ ∂p ⎠T
⎦
but a function of temperature alone.
Since for an ideal gas, pV = nRT
and
⎛ ∂V ⎞
V −T ⎜
⎟ =0
⎝ ∂T ⎠ p
the relation dH = Cp dT is true for any process (even when the pressure changes.)
However, for any other substance the relation dH = Cp dT holds good only when the pressure
remains constant or dp = 0.
(b) Thermal radiation in equilibrium with the enclosing walls processes an energy that depends
only on the volume and temperature. The energy density (u), defined as the ratio of energy to
volume, is a function of temperature only, or
u=
U
= f (T )only.
V
The electromagnetic theory of radiation states that radiation is equivalent to a photon gas and
it exerts a pressure, and that the pressure exerted by the black body radiation in an enclosure
is given by
p=
u
3
Black body radiation is thus specified by the pressure, volume and temperature of the
radiation.
since.
u
3
1 du
⎛ ∂U ⎞
⎛ ∂p ⎞
⎜ ∂V ⎟ = u and ⎜ ∂T ⎟ = 3 dT
⎝
⎠T
⎝
⎠V
U = uV and p =
By substituting in the energy Eq.
T du u
−
3 dT 3
du
dT
∴
=4
u
T
u=
Page 190 of 265
Thermodynamic Relations
Chapter 11
or
ln u = ln T4 + lnb
or
u = bT4
where b is a constant. This is known as the Stefan - Boltzmann Law.
Since U = uV = VbT 4
and
⎛ ∂U ⎞
3
⎜ ∂T ⎟ = Cv = 4VbT
⎝
⎠V
1 du 4
⎛ ∂p ⎞
3
⎜ ∂T ⎟ = 3 dT = 3 bT
⎝
⎠V
From the first TdS equation
⎛ ∂p ⎞
TdS = Cv dT + T ⎜
⎟ dV
⎝ ∂T ⎠v
4
= 4VbT 3 dT + bT 4 .dV
3
For a reversible isothermal change of volume, the heat to be supplied reversibly to keep
temperature constant.
Q=
4
bT 4 ΔV
3
For a reversible adiabatic change of volume
4
bT 4 dV = −4VbT 3 dT
3
dV
dT
or
= −3
V
T
3
or VT = const
If the temperature is one-half the original temperature. The volume of black body radiation is to
be increased adiabatically eight times its original volume so that the radiation remains in
equilibrium with matter at that temperature.
Gibbs Phase Rule
Gibbs Phase Rule determines what is expected to define the state of a system
F=C–P+2
F = Number of degrees of freedom (i.e.., no. of properties required)
C = Number of components
P = Number of phases
e.g., Nitrogen gas C = 1; P = 1. Therefore, F = 2
•
•
•
•
•
To determine the state of the nitrogen gas in a cylinder two properties are adequate.
A closed vessel containing water and steam in equilibrium: P = 2, C = 1
Therefore, F = 1. If any one property is specified it is sufficient.
A vessel containing water, ice and steam in equilibrium
P = 3, C = 1 therefore F = 0. The triple point is uniquely defined.
Question: Which one of the following can be considered as property of a system?
(a) ∫ pdv
(b) ∫ vdp
⎛ dT p.dv ⎞
(c ) ∫ ⎜
+
⎟
v ⎠
⎝ T
⎛ dT v.dp ⎞
(d ) ∫ ⎜
−
⎟
T ⎠
⎝ T
Given: p = pressure, T = Temperature, v = specific volume
Page 191 of 265
[IES-1993]
Thermodynamic Relations
Chapter 11
Solution: P is a function of v and both are connected by a line path on p and v coordinates.
Thus
∫ pdv and ∫ vdp are not exact differentials and thus not properties.
If X and Y are two properties of a system, then dx and dy are exact differentials. If
the differential is of the form Mdx + Ndy, then the test for exactness is
⎡ ∂M ⎤ ⎡ ∂N ⎤
⎢ ∂y ⎥ = ⎢ ∂x ⎥
⎣
⎦x ⎣ ⎦ y
Now applying above test for
2
R
⎛ dT p.dv ⎞ ⎡ ∂ (1/ T ) ⎤
⎡ ∂ ( p / v) ⎤ ⎡ ∂ ( RT / v ) ⎤
∫ ⎜⎝ T + v ⎟⎠ , ⎢⎣ ∂v ⎥⎦T = ⎢⎣ ∂T ⎥⎦ v = ⎢⎣ ∂T ⎥⎦ or 0 = v 2
v
This differential is not exact and hence is not a point function and hence
⎛ dT
p.dv ⎞
⎟ is not a point function and hence not a property.
v ⎠
⎛ dT v.dp ⎞ ⎡ ∂ (1/ T ) ⎤
⎡ ∂ ( −v / T ) ⎤
⎡ ∂ (− R / P) ⎤
=⎢
=⎢
And for ∫ ⎜
−
⎟⎢
⎥
⎥
⎥⎦ or 0 = 0
T ⎠ ⎣ ∂p ⎦T ⎣ ∂T ⎦ P ⎣ ∂T
⎝ T
P
∫ ⎜⎝ T
+
Thus
∫ ⎜⎝ T
⎛ dT
−
v.dp ⎞
⎟ is exact and may be written as ds, where s is a point function and
T ⎠
hence a property
Page 192 of 265
Vapour Power Cycles
Chapter 12
12. Vapour Power Cycles
Some Important Notes
A.
Rankine Cycle
Q1
T
4
WP
WT
3
Q2
h 1 = W T + h2
WT = h1 − h2
(iii)
or
h3 + W P = h4
WP = h4 − h3
2
p2
S
For 1 kg of fluid using S.F.E.E.
h4 + Q1 = h1
(i)
Q1 = h1 − h4
or
(ii)
or
1
About pump: The pump handles liquid water which is incompressible.
For reversible Adiabatic Compression Tds = dh – vdp where ds = 0
∴
dh = vdp as v = constant
Δh = vΔp
or
h 4 − h3 = v(p1 − p2 ) = WP
(iv)
B.
WP = h4 − h3 = v(p1 − p2 ) kJ/kg
Where v in m3 /kg and p in kPa
Rankine Cycle efficiency:
η=
Wnet
(h1 − h 2 ) − (h 4 − h3 )
W − NP
= T
=
Q1
(h1 − h 4 )
Q1
3600
kg
WT − WP kWh
C.
Steam rate =
D.
Heat Rate = Steam rate × Q1 =
3600 Q1 kJ
3600 kJ
=
WT − WP kWh
η kWh
Page 193 of 265
Vapour Power Cycles
Chapter 12
E.
About Turbine Losses: If there is heat loss to the surroundings, h2 will decrease,
accompanied by a decrease in entropy. If the heat loss is large, the end state of steam
from the turbine may be 2′.(figure in below).
It may so happen that the entropy increase due to frictional effects just balances the
entropy decrease due to heat loss, with the result that the initial and final entropies of
steam in the expansion process are equal, but the expansion is neither adiabatic nor
reversible.
F.
Isentropic Efficiency:
ηisen =
h1 − h 2
h1 − h 2s
=
Actual Enthalpy drop
isentropic enthalpy drop
Q1
T↑
1
4
Wp
WT
3
Q2
2′
2s 2
S→
G.
Mean temperature of heat addition:
Q1 = h1 − h 4 s = Tm (s1 − s4 s )
∴
Tm =
h1 − h 4 s
s1 − s4 s
1
5
T
↑
Tm
4s
2s
3
→S
Page 194 of 265
Vapour Power Cycles
Chapter 12
H.
For Reheat – Regenerative Cycle:
1
1 kg
2
12
11
10
T
m1 kg
9
8
4
(1–m1 )kg
5
3
(1–m1 –m2 )kg
m2 kg
7
6
(1–m1 –m2 )kg
S
WT = (h1 – h2) + (1 – m1) (h2 – h3) + (1 – m1) (h4 – h5) + (1 – m1 – m2) (h5 – h6) kJ/kg
WP = (1 – m1 – m2) (h8 – h7) + (1 – m1) (h10 – h9) + 1(h12 – h11) kJ/kg
Q1 = (h1 – h12) + (1 – m1) (h4 – h3) kJ/kg
Energy balance of heater 1 and 2
m1 h2 + (1 – m1) h10 = 1 × h11 ………… For calculation of m1
And m2 h5 + (1 – m1 – m2) h8 = (1 – m1 ) h9 ……... For calculation of m2 .
I.
For Binary vapour Cycles:
m kg
a
1
d
b
c
T
5
1 kg 6
4
3
2
S
WT = m (ha – hb) + (h1 – h2) kJ/kg of steam
WP = m (hd – hc) + (h4 – h3) kJ /kg of steam
Q1 = m (ha – hd) + (h1 – h6) + (h5 – h4) kJ /kg of steam.
Energy balance in mercury condenser-steam boiler
m (hb – hc) = (h6 – h5)
h − h5
kg of Hg/kg of H2O i.e. ≈ 8 kg
∴
m= 6
hb − hc
Page 195 of 265
Vapour Power Cycles
Chapter 12
J.
Efficiency of Binary vapour cycle:
1 – η = (1 − η1 ) (1 − η2 ) ........ (1 − ηn )
∴
For two cycles
η = n1 + n 2 − n1 n 2
K.
Overall efficiency of a power plant
ηoverall = ηboiler × ηcycle × ηturbine (mean) × ηgenerator
Questions with Solution P. K. Nag
Q. 12.1
for the following steam cycles find
(a) WT in kJ/kg
(b) Wp in kJ/kg,
(d) cycle efficiency,
(c) Q1 in kJ/kg,
(e) steam rate in kg/kW h, and
(f) moisture at the end of the turbine process. Show the results in
tabular form with your comments.
Boiler Outlet
Type of Cycle
Condenser Pressure
10 bar, saturated 1 bar
Ideal Rankine Cycle
-do-
-do-
Neglect Wp
-do-
-do-
-do-
0.1 bar
Assume 75% pump and
Turbine efficiency
Ideal Rankine Cycle
10 bar, 300°c
-do-
-do-
150 bar, 600°c
-do-
-do-
-do-
-do-
-do-
-do-
Reheat to 600°C
maximum intermediate
pressure to limit end
moisture to 15%
-do- but with 85% tur- bine
efficiencies
Isentropic pump process ends
on satura
Type of Cycle
10 bar, saturated 0.1 bar
Boiler Outlet
Condenser Pressure
10 bar, saturated 0.1 bar
-do-
-do-
-do-
-do-
-do-
-do-
-do-
-do-
-do- but with 80% machine
efficiencies
Ideal regenerative cycle
Single open heater at
110°c
Two open heaters at 90°c
and 135°c
-do- but the heaters are
closed heaters
Page 196 of 265
Vapour Power Cycles
Chapter 12
Solution:
Boiler outlet: 10 bar, saturated
Condenser: 1 bar
Ideal Rankine Cycle
p = 10 bar
T
1
4
p = 1 bar
3
From Steam Table
h1 = 2778.1 kJ/kg
2
S
s1 = 6.5865 kJ/kg-K
∴
s2 = s1 = 6.5865 = 1.3026 + x (7.3594 – 1.3026)
∴
∴
x = 0.8724
h 2 = 417.46 + 0.8724 × 2258 = 2387.3 kJ/kg
h3 = 417.46 kJ/kg
∴
(a)
(b)
(c)
(d)
(e)
(f)
h4 = h3 + WP
WP = 1.043 × 10–3 [1000 – 100] kJ/kg = 0.94 kJ/kg
h4 = 418.4 kJ/kg
WT = h1 – h2 = (2778.1 – 2387.3) kJ/kg = 390.8 kJ/kg
WP = 0.94 kJ/kg
Q1 = (h1 – h4) = (2778.1 – 418.4) kJ/kg = 2359.7 kJ/kg
W
W − NP
390.8 − 0.94
Cycle efficiency (η) = net = T
=
2359.7
Q1
Q1
= 16.52%
3600
3600
Steam rate =
kJ / kWh =
= 9.234 kg/kWh
Wnet
390.8 − 0.94
Moisture at the end of turbine process
= (1 – x) = 0.1276 ≅ 12.76%
Q.12.2
A geothermal power plant utilizes steam produced by natural means
underground. Steam wells are drilled to tap this steam supply which is
available at 4.5 bar and 175°C. The steam leaves the turbine at 100 mm
Hg absolute pressure. The turbine isentropic efficiency is 0.75. Calculate
the efficiency of the plant. If the unit produces 12.5 MW, what is the
steam flow rate?
Solution:
p1 = 4.5 bar
T1 = 175ºC
From super heated STEAM TABLE.
Page 197 of 265
Vapour Power Cycles
Chapter 12
p1
1
T
p2
2
2′
S
At 4 bar
150°C
h = 2752.8
s = 6.9299
200°C
h = 2860.5
s = 7.1706
at 5 bar
152°C
h = 2748.7
s = 6.8213
200°C
h = 2855.4
s = 7.0592
∴
at 4 bar 175°C
at 5 bar, 175°C
1
h = 2752.8 + (2860.5 − 2752.8)
2
⎛ 175 − 152 ⎞
h = 2748.7 + ⎜
⎟ (2855.4 − 2748.7)
⎝ 200 − 152 ⎠
= 2806.7 kJ/kg
= 2800 kJ/kg
1
23
s = 6.9299 + (7.1706 − 6.9299) s = 6.8213 + (7.0592 − 6.8213)
2
48
= 7.0503 kJ/kg – K
= 6.9353
∴
at 4.5 bar 175°C
2806.7 + 2800
h1 =
= 2803.4 kJ/kg
2
7.0503 + 6.9353
s1 =
= 6.9928 kJ/kg – K
2
Pressure 100 mm Hg
100
=
m × (13.6 × 103 ) kg / m3 × 9.81 m/s2
1000
= 0.13342 bar = 13.342 kPa
Here also entropy 6.9928 kJ/kg – K
So from S. T.
At 10 kPa
at 15 kPa
hf = 191.83
sf = 0.6493 sf = 0.7549 hf = 225.94
hfg = 2392.8
sg = 8.1502
sg = 8.0085
hfg = 2373.1
∴
at 13.342 kPa [Interpolation]
⎛ 15 − 13.342 ⎞
sf = 0.6493 + ⎜
⎟ (0.7549 – 0.6493) = 0.68432 kJ/kg – K
⎝ 15 − 10 ⎠
⎛ 15 − 13.342 ⎞
⎟ (8.0085 – 8.1502) = 8.1032 kJ/kg – K
⎠
If dryness fraction is x then
6.9928 = 0.68432 + x (8.1032 – 0.68432)
x = 0.85033
At 13.342 kPa
Page 198 of 265
sg = 8.1502 + ⎜
⎝ 15 − 10
∴
∴
∴
Vapour Power Cycles
Chapter 12
⎛ 15 − 13.342 ⎞
h f = 191.83 + ⎜
⎟ (225.94 – 191.83) = 203.14 kJ/kg
⎝ 15 − 10 ⎠
⎛ 15 − 13.342 ⎞
h fg = 2392.8 + ⎜
⎟ (2373.1 – 2392.8) = 2386.3 kJ/kg
⎝ 15 − 10 ⎠
h2s = hf + x hfg = 203.14 + 0.85033 × 2386.3 = 2232.3 kJ/kg
ηisentropic =
∴
h1 − h 2′
h1 − h 2s
h1 − h 2′ = ηisentropic × (h1 – h2s)
h′2 = h1 – ηisentropic (h1 – h2s)
∴
= 2803.4 – 0.75 (2803.4 – 2232.3) = 2375 kJ/kg.
Turbine work (WT) = h1 − h 2′ = (2803.4 – 2373) %
∴
= 428.36 kJ/kg
W
428.36
≈ 0.1528 = 25.28%
∴
Efficiency of the plant = T =
2803.4
h1
•
If mass flow rate is m kg/s
•
m. WT = 12.5 × 103
or
•
m =
12.5 × 103
= 29.18 kg/s
428.36
Q.12.3
A simple steam power cycle uses solar energy for the heat input. Water
in the cycle enters the pump as a saturated liquid at 40°C, and is pumped
to 2 bar. It then evaporates in the boiler at this pressure, and enters the
turbine as saturated vapour. At the turbine exhaust the conditions are
40°C and 10% moisture. The flow rate is 150 kg/h. Determine (a) the
turbine isentropic efficiency, (b) the net work output (c) the cycle
efficiency, and (d) the area of solar collector needed if the collectors pick
up 0.58 kW/ m 2 .
(Ans. (c) 2.78%, (d) 18.2 m 2 )
Solution:
From Steam Table
T1 = 120.23°C = 393.23 K
h1 = 2706.7 kJ/kg
s1 = 7.1271 kJ/kg – K
2 bar
1
T
4
2s 2
3
At 40°C saturated pressure 7.384 kPa
hf = 167.57
S
hfg = 2406.7
Page 199 of 265
Vapour Power Cycles
Chapter 12
sf = 0.5725
sg = 8.2570
∴
h2 = hf + 0.9 × 2406.7 = 2333.6 kJ/kg
For h2s if there is dryness fraction x
7.1271 = 0.5725 + x × (8.2570 – 0.5725)
∴
x = 0.853
∴
h2s = 167.57 + 0.853 × 2406.7 = 2220.4 kJ/kg
h − h2
(a)
∴ Isentropic efficiency, ηisentropic = 1
h1 − h 2s
2706.7 − 2333.6
=
= 76.72%
2706.7 − 2220.4
(b)
Net work output WT = h1 – h2 = 373.1 kJ/kg
∴
•
Power = 15.55 kW i.e. (WT − WP ) × m
Pump work, WP = v ( p1 – p2 )
∴
∴
∴
= 1.008 × 10–3 (200 – 7.384) kJ/kg = 0.1942 kJ/kg
h3 = 167.57 kJ/kg,
ha = 167.76 kJ/kg
Q1 = (h1 – h4) = (2706.7 – 167.76) kJ/kg = 2539 kJ/kg
W − WP
373.1 − 0.1942
ηcycle = T
=
= 14.69 %
2539
Q1
•
Q1 × m
Required area A =
collection picup
2539 × 150
= 182.4 m2
=
0.58 × 3600
Q.12.4
Solution :
In a reheat cycle, the initial steam pressure and the maximum
temperature are 150 bar and 550°C respectively. If the condenser
pressure is 0.1 bar and the moisture at the condenser inlet is 5%, and
assuming ideal processes, determine (a) the reheat pressure, (b) the
cycle efficiency, and (c) the steam rate.
(Ans. 13.5 bar, 43.6%,2.05 kg/kW h)
From Steam Table at 150 bar 550°C
h1 = 3448.6 kJ/kg
s1 = 6.520 kJ/kg – K
At
p3 = 0.1 bar
T = 45.8°C
h f = 191.8 kJ/kg
h fg = 2392.8 kJ/kg
Page 200 of 265
Vapour Power Cycles
Chapter 12
p1
1
T
p2
3
2
6
5
∴
p3
4
S
h4 = hf + x hfg = 191.8 + 0.95 × 2392.8 = 2465 kJ/kg
sf = 0.649: Sfg = 7.501
s4 = sf + x sfg = 0.649 + 0.95 × 7.501 = 7.775 kJ/kg – K
From Molier Diagram at 550°C and 7.775 entropy, 13.25 bar
From S.T. at 10 bar 550°C
s = 7.8955
∴
15 bar 550°C
s = 7.7045
⎛ p − 10 ⎞
∴
7.775 = 7.8955 + ⎜
⎟ (7.7045 − 7.8955)
⎝ 15 − 10 ⎠
–0.1205 = (p – 10) (–0.0382)
∴
p – 10 = 3.1544 ⇒ p = 13.15 bar
∴
from Molier Dia. At 13 bar 550°C
h3 = 3580 kJ/kg
t2 = 195°C
h2 = 2795 kJ/kg
h5 = 191.8 kJ/kg
WP = v 5 ( p1 – p3 ) = 0.001010 (15000 – 10) kJ/kg
∴
∴
∴
∴
= 1.505 kJ/kg
h6 = h5 + WP = 193.3 kJ/kg
WT = (h1 – h2) + (h3 – h4) = 1768.6 kJ/kg
WP = 1.50 kJ/kg
Wnet = 1767.5 kJ/kg
Q = (h1 – h6) + (h3 – h2) = 4040.3 kJ/kg
W
1767.5
ηcycle = net =
× 100 % = 43.75%
4040.3
Q
Steam rate =
Q.12.5
3600
3600
kg / kWh = 2.0368 kg/kWh
=
1767.5
WT − WP
In a nuclear power-plant heat is transferred in the reactor to liquid
sodium. The liquid sodium is then pumped to a heat exchanger where
heat is transferred to steam. The steam leaves this heat exchanger as
saturated vapour at 55 bar, and is then superheated in an external gasfired super heater to 650°C. The steam then enters the turbine, which
has one extraction point at 4 bar, where steam flows to an open feed
water heater. The turbine efficiency is 75% and the condenser
temperature is 40°C. Determine the heat transfer in the reactor and in
the super heater to produce a power output of 80 MW.
Page 201 of 265
Vapour Power Cycles
Chapter 12
Solution:
From Steam Table at 55 bar saturated state
h9 = 2789.9 kJ/kg
ga
Na
7
T
8
6
s
1
1 kg
9
2
m kg
(1–m) kg
5
4
(1–m)kg
3
S
From super heated S.T. at 55 bar 650°C
at 50 bar 600°C,
700°C
∴ By calculation at 650°C
h = 3666.5
h = 3900.1
h = 3783.3
s = 7.2589
s = 7.5122
s = 7.3856
At 60 bar 600°
h = 3658.4
s = 7.1677
C = 700°C
h = 3894.2
s = 7.4234
∴ by calculation
h = 3776.3
s = 7.2956
∴
at 55 bar 650°C (by interpolation)
h1 = 3770.8 kJ/kg
s1 = 7.3406 kJ/kg
For h2, at 4 bar where S = 7.3406
At 200°C
at 250°C if temp is t
s = 7.172,
s = 7.379
h = 2860.5
h = 2964.2
⎛ 7.3406 − 7.171 ⎞
Then h2 = 2860.5 + ⎜
⎟ × (2964.2 − 2860.5) = 2945 kJ/kg
⎝ 7.379 − 7.171 ⎠
For h3, at point 3 at 40°C
hfg = 2406.7
hf = 167.6,
sf = 0.573
sfg = 7.685
If dryness fraction is x then
0.573 + x × 7.685 = 7.3406
∴
x = 0.8806
∴
h3 = 167.6 + 0.8806 × 2406.7 = 2287 kJ/kg
h4 = hf = 167.6 kJ/kg
WP4 – 5 = v 4 ( p2 – p3 ) = 0.001010 × (400 – 7.38) kJ/kg
= 0.397 kJ/kg ≈ 0.4 kJ/kg
∴ h5 = h 4 + WP4 − 5 = 168 kJ/kg
h6 = 604.7 kJ/kg
[at 4 bar saturated liquid]
Page 202 of 265
Vapour Power Cycles
Chapter 12
WP6 − 7 = v 6 ( p1 – p2 ) = 0.001084 (5500 – 400) = 5.53 kJ/kg
∴
h7 = h 6 + WP6 − 7 = 610.23 kJ/kg
From heater energy balance
⇒ m = 0.1622 kg
(1 – m) h5 + mh2 = h5
∴
WT = [(h1 – h2) + (1 – m) (h2 – h3) × 0.75 = 1049.8 kJ/kg ;
Wnet = WT − WP4 − 5 − WP6 − 7 = 1043.9 kJ/kg
•
80 × 103
kg / s = 76.638 kg/s
1049.8
∴
Steam flow rate (m) =
∴
Heat transfer in heater = m(h 9 − h7 )
•
= 76.638(2789.9 – 610.23) = 167.046 MW
•
Heat transfer in super heater = m(h1 − h 9 )
= 76.638(3779.8 – 2789.9) = 75.864 MW
Q.12.6
Solution:
Q.12.7
Solution:
Q.12.8
Solution:
Q.12.9
In a reheat cycle, steam at 500°C expands in a h.p. turbine till it is
saturated vapour. It is reheated at constant pressure to 400°C and then
expands in a l.p. turbine to 40°C. If the maximum moisture content at the
turbine exhaust is limited to 15%, find (a) the reheat pressure, (b) the
pressure of steam at the inlet to the h.p. turbine, (c) the net specific
work output, (d) the cycle efficiency, and (e) the steam rate. Assume all
ideal processes.
What would have been the quality, the work output, and the cycle
efficiency without the reheating of steam? Assume that the other
conditions remain the same.
Try please.
A regenerative cycle operates with steam supplied at 30 bar and 300°C
and -condenser pressure of 0.08 bar. The extraction points for two
heaters (one Closed and one open) are at 3.5 bar and 0.7 bar respectively.
Calculate the thermal efficiency of the plant, neglecting pump work.
Try please.
The net power output of the turbine in an ideal reheat-regenertive cycle
is 100 MW. Steam enters the high-pressure (H.P.) turbine at 90 bar,
550°C. After .expansion to 7 bar, some of the steam goes to an open
heater and the balance is reheated to 400°C, after which it expands to
0.07 bar. (a) What is the steam flow rate to the H.P. turbine? (b) What is
the total pump work? (c) Calculate the cycle efficiency. (d) If there is a
10°c rise in the temperature of the cooling 'water, what is the rate of
flow of the cooling water in the condenser? (e) If the velocity of the
steam flowing from the turbine to the condenser is limited to a
maximum of 130 m/s, find the diameter of the connecting pipe.
Try please.
A mercury cycle is superposed on the steam cycle operating between the
boiler outlet condition of 40 bar, 400°C and the condenser temperature
Page 203 of 265
Vapour Power Cycles
Chapter 12
of 40°C. The heat released by mercury condensing at 0.2 bar is used to
impart the latent heat of vaporization to the water in the steam cycle.
Mercury enters the mercury turbine as saturated vapour at 10 bar.
Compute (a) kg of mercury circulated per kg of water, and (b) the
efficiency of the combined cycle.
The property values of saturated mercury are given below
p
T( °C )
hf (kJ/kg)
s f (kJ/kg k)
vf ( m3 /kg)
(bar)
hg
sg
vg
10
515.5
72.23
363.0
0.1478
0.5167
0.2
277.3
38.35
336.55
0.0967
0.6385
80.9 x 10−6
0.0333
77.4 x 10−6
1.163
Solution:
Try please.
Q.12.10
In an electric generating station, using a binary vapour cycle with
mercury in the upper cycle and steam in the lower, the ratio of mercury
flow to steam flow is 10 : 1 on a mass basis. At an evaporation rate of
1,000,000 kg/h for the mercury, its specific enthalpy rises by 356 kJ/kg in
passing through the boiler. Superheating the steam in the boiler furnace
adds 586 kJ to the steam specific enthalpy. The mercury gives up 251.2
kJ/kg during condensation, and the steam gives up 2003 kJ/kg in its
condenser. The overall boiler efficiency is 85%. The combined turbine
metrical and generator efficiencies are each 95% for the mercury and
steam units. The steam auxiliaries require 5% of the energy generated by
the units. Find the overall efficiency of the plant.
Try please
Solution:
Q.12.11
A sodium-mercury-steam cycle operates between l000°C and 40°C.
Sodium rejects heat at 670°C to mercury. Mercury boils at 24.6 bar and
rejects heat at 0.141 bar. Both the sodium and mercury cycles are
saturated. Steam is formed at 30 bar and is superheated in the sodium
boiler to 350°C. It rejects heat at 0.0 8 bar. Assume isentropic expansions,
no heat losses, and no generation and neglect pumping work. Find (a)
the amounts of sodium and mercury used per kg of steam, (b) the heat
added and rejected in the composite cycle per kg steam, (c) the total
work done per kg steam. (d) the efficiency of the composite cycle, (e) the
efficiency of the corresponding Carnot cycle, and (f) the work, heat
added, and efficiency of a supercritical pressure steam (single fluid)
cycle operating at 250 bar and between the same temperature limits.
For mercury, at 24.6 bar, hg = 366.78 kJ/kg
sg = 0.48kJ/kg K
And
and at 0.141 bar, s j =0.09
sg = 0.64kJ/kg K, h j =36.01 and h g =330.77 kJ/kg
For sodium, at 1000°C, hg = 4982.53 kJ/kg
At turbine exhaust = 3914.85 kJ/kg
At 670°C, hf = 745.29 kJ/kg
Solution:
For a supercritical steam cycle, the specific enthalpy and entropy at the
turbine inlet may be computed by extrapolation from the steam tables.
Try please.
Page 204 of 265
Vapour Power Cycles
Chapter 12
Q.12.12
Solution:
Q.12.13
Solution:
Q.12.14
Solution:
Q.12.15
Solution:
A textile factory requires 10,000 kg/h of steam for process heating at 3
bar saturated and 1000 kW of power, for which a back pressure turbine
of 70% internal efficiency is to be used. Find the steam condition
required at the inlet to the turbine.
Try please.
A 10,000 kW steam turbine operates with steam at the inlet at 40 bar,
400°C and exhausts at 0.1 bar. Ten thousand kg/h of steam at 3 bar are to
be extracted for process work. The turbine has 75% isentropic efficiency
throughout. Find the boiler capacity required.
Try please.
A 50 MW steam plant built in 1935 operates with steam at the inlet at 60
bar, 450°C and exhausts at 0.1 bar, with 80% turbine efficiency. It is
proposed to scrap the old boiler and put in a new boiler and a topping
turbine of efficiency 85% operating with inlet steam at 180 bar, 500°C.
The exhaust from the topping turbine at 60 bar is reheated to 450°C and
admitted to the old turbine. The flow rate is just sufficient to produce
the rated output from the old turbine. Find the improvement in
efficiency with the new set up. What is the additional power developed?
Try please.
A steam plant operates with an initial pressure at 20 bar and
temperature 400°C, and exhausts to a heating system at 2 bar. The
condensate from the heating system is returned to the boiler plant at
65°C, and the heating system utilizes for its intended purpose 90% of the
energy transferred from the steam it receives. The turbine efficiency is
70%. (a) What fraction of the energy supplied to the steam plant serves a
useful purpose? (b) If two separate steam plants had been set up to
produce the same useful energy, one to generate heating steam at 2 bar,
and the other to generate power through a cycle working between 20
bar, 400°C and 0.07 bar, what fraction of the energy supplied would have
served a useful purpose?
(Ans. 91.2%, 64.5%)
From S.T. at 20 bar 400°C
h1 = 3247.6 kJ/kg
s1 = 7.127 kJ/kg – K
1
20 bar
T
3
65°C
At 2 bar
2 bar
4
Q0
2
S
Page 205 of 265
Vapour Power Cycles
Chapter 12
sf = 1.5301, sfg = 5.5967
sg = 7.127 kJ/kg – K so at point (2)
Steam is saturated vapour
So
h2 = 2706.3 kJ/kg
At 2 bar saturated temperature is 120.2°C but 65°C liquid
So
h3 = h2 – CP ΔT
= 504.7 – 4.187 × (120.2 – 65) = 273.6 kJ/kg
WP3 − 4 = v 3 ( p1 – p2 ) = 0.001 × (2000 – 200) = 1.8 kJ/kg
∴ h4 = 275.4 kJ/kg
∴ Heat input (Q) = h1 – h4 = (3247.6 – 275.4) = 2972.2 kJ/kg
Turbine work = (h1 – h2) η = (3247.6 – 2706.3) × 0.7 kJ/kg
= 378.9 kJ/kg
Heat rejection that utilized (Q0) = (h2 – h3) η
= (2706.3 – 273.6) × 0.9 = 2189.4 kJ/kg
∴ Net work output (Wnet) = WT – WP = 378.9 – 1.8
= 377.1 kJ/kg
∴ Fraction at energy supplied utilized
Wnet + Q0
377.1 + 2189.4
=
× 100%
2972.2
Q1
= 86.35%
=
(b) At 0.07 bar
sf = 0.559, sfg = 7.717
∴ Dryness fraction x, 0.559 + x × 7.717 = 7.127
∴
x = 0.85137
∴
h2 = 163.4 + 0.85137 × 2409.1 = 2214.4 kJ/kg
h3 = 163.4 kJ/kg
∴
∴
WP = 0.001007 × (2000 – 7) = 2 kJ/kg
h4 = 165.4 kJ/g.
WT = (h1 – h2) × 0.7 = 723.24 kJ/kg
Wnet = WT – WP = 721.24 kJ/kg
Here heat input for power = (h1 – h4) = 3082.2. kJ/kg
For same 377.1 kg power we need 0.52285 kg of water
So heat input = 1611.5 kJ for power
2189.4
kJ = 2432.7 kJ
Heat input for heating =
0.9
377.1 + 2189.4
× 100%
∴ Fraction of energy used =
1611.5 + 2432.7
= 63.46%
Q.12.16
In a nuclear power plant saturated steam at 30 bar enters a h.p. turbine
and expands isentropically to a pressure at which its quality is 0.841. At
this pressure the steam is passed through a moisture separator which
removes all the liquid. Saturated vapour leaves the separator and is
Page 206 of 265
Vapour Power Cycles
Chapter 12
Solution:
expanded isentropically to 0.04 bar in I.p. turbine, while the saturated
liquid leaving the separator is returned via a feed pump to the boiler.
The condensate leaving the condenser at 0.04 bar is also returned to the
boiler via a second feed pump. Calculate the cycle efficiency and turbine
outlet quality taking into account the feed pump term. Recalculate the
same quantities for a cycle with the same boiler and condenser
pressures but without moisture separation.
(Ans. 35.5%, 0.S24; 35%; 0.716)
Form Steam Table at 30 bar saturated
h1 = 2802.3 kJ/kg
s1 = 6.1837
From Molier diagram
h2 = 2300 kJ/kg
pr = 2.8 bar
From S.T. hg =2721.5 kJ/kg, sg = 7.014 kJ/kg K
t = 131.2°C
hf = 551.4 kJ/kg
From 0.04 bar S.T
sf = 0.423 kJ/kg, sfg = 8.052 kJ/kg
hf = 121.5 kJ/kg, hfg = 2432.9 kJ/kg
1
8
T
30 bar
1 kg
3
6
7
5
p
m kg 2
(1–m)kg
(1–m)kg
0.7 bar
4
S
∴
If dryness fraction is x the
7.014 = 0.423 + x × 8.052
⇒
x = 0.8186
∴
h4 = hf + x hfg = 2113 kJ/kg
WP5– 6 = 0.001(3000 – 4) = 3 kJ/kg
WP7 – 8 = 0.001071(3000 – 280) = 2.9 kJ/kg
So
h1 – 2802.3 kJ/kg
h2 – 2380 kJ/kg
h3 – 2721.5 kJ/kg
h4 – 2113 kJ/kg
∴
∴
h5 = 121.5 kJ/kg
h6 = 124.5 kJ/kg
h7 = 551.4 kJ/kg
h8 = 554.3 kJ/kg
m = 1 – 0. 841 = 0.159 kg of sub/ kg of steam
WT = (h1 – h2) + (1 – m) (h3 – h4) = 934 kJ/kg
WP = m × WP7 − 8 + (1 – m) WP5– 6 = 2.98 kJ/kg ≈ 3 kJ/kg
Page 207 of 265
Vapour Power Cycles
Chapter 12
∴ Wnet = 931 kJ/kg
Heat supplied (Q) = m(h1– h8) + (1 – m) (h1 – h6) = 2609.5 kJ/kg
931
× 100% = 35.68% with turbine exhaust quality 0.8186
2609.5
If No separation is taking place, Then is quality of exhaust is x
⇒ x = 0.715
Then 6.1837 = 0.423 + x × 8.052
∴η=
∴
h4 = hf + x × hfg = 1862 kJ/kg
∴
WT = h1-h4 = 941.28 kJ/kg
WP = WP5 − 6 = 3 kJ/kg
∴
Wnet = 938.28 kJ/kg
∴
Heat input, Q = h1 – h6 = 2677.8 kJ/kg
938.28
∴
η=
× 100% = 35%
2677.8
Q.12.17
Solution:
The net power output of an ideal regenerative-reheat steam cycle is
80MW. Steam enters the h.p. turbine at 80 bar, 500°C and expands till it
becomes saturated vapour. Some of the steam then goes to an open
feedwater heater and the balance is reheated to 400°C, after which it
expands in the I.p. turbine to 0.07 bar. Compute (a) the reheat pressure,
(b) the steam flow rate to the h.p. turbine, and (c) the cycle efficiency.
Neglect pump work.
(Ans. 6.5 bar, 58.4 kg/s, 43.7%)
From S.T of 80 bar 500°C
h1 = 3398.3 kJ/kg
s1 = 6.724 kJ/kg – K
s2 = 6.725 at 6.6 bar so
Reheat pr. 6.6 bar
1 80 bar 500°C
3
400°C
8
T
7
6
m kg
(1 – m) kg
2
0.07 bar
5
(1 – m) kg
4
S
∴
h2 = 2759.5 kJ/kg
h3 = 3270.3 + 0.6(3268.7 – 3270.3) = 3269.3 kJ/kg
s3 = 7.708 + 0.6 (7.635 – 7.708)
= 7.6643 kJ/kg – K
At 0.07 bar
hf = 163.4,
hfg = 2409.1
Page 208 of 265
Vapour Power Cycles
Chapter 12
hf = 0.559,
sfg = 7.717
∴ If quality is x then
7.6642 = 0.559 + x × 7.717 ⇒ x = 0.9207
∴ h4 = 163.4 + 0.9207 × 2409.1 = 2381.5 kJ/kg
h7 = 686.8 kJ/kg ≈ h8
h5 = 163.4 kJ/kg ≈ h6,
∴ From Heat balance of heater
m × h2 + (1 – m) h6 = h7
∴ m = 0.2016 kg/kg of steam at H.P
∴ (1 – m) = 0.7984
WT = h1 – h2 + (1 – m) (h3 –h4) = 1347.6kJ/kg
WP neglected
Q = (h1 – h8) + (1 – m) (h3 – h2) = 3118.5 kJ/kg at H.P
∴ (a) Reheat pr. 6.6 bar
(b) Steam flow rate at H.P =
(c) Cycle efficiency (η) =
Q.12.18
80 × 103
kg/s = 59.36 kg/s
1347.6
W
1347.6
=
× 100% = 43.21%
3118.5
Q
Figure shows the arrangement of a steam plant in which steam is also
required for an industrial heating process. The steam leaves boiler B at
30 bar, 320°C and expands in the H.P. turbine to 2 bar, the efficiency of
the H.P. turbine being 75%. At this point one half of the steam passes to
the process heater P and the remainder enters separator S which
removes all the moisture. The dry steam enters the L.P. turbine at 2 bar
and expands to the condenser pressure 0.07 bar, the efficiency of the L.P.
turbine being 70%. The drainage from the
Page 209 of 265
Vapour Power Cycles
Chapter 12
Solution:
Separator mixes with the condensate from the process heater and the
combined flow enters the hotwell H at 50°C. Traps are provided at the
exist from P and S. A pump extracts the condensate from condenser C
and this enters the hot well at 38°C. Neglecting the feed pump work and
radiation loss, estimate the temperature of water leaving the hotwell
which is at atmospheric pressure. Also calculate, as percentage of heat
transferred in the boiler, (a) the heat transferred in the process heater,
and (b) the work done in the turbines.
Try please.
Q.12.19
In a combined power and process plant the boiler generates 21,000 kg/h
of steam at a pressure of 17 bar, and temperature 230 °C . A part of the
steam goes to a process heater which consumes 132.56 kW, the steam
leaving the process heater 0.957 dry at 17 bar being throttled to 3.5 bar.
The remaining steam flows through a H.P. turbine which exhausts at a
pressure of 3.5 bar. The exhaust steam mixes with the process steam
before entering the L.P. turbine which develops 1337.5 kW. At the
exhaust the pressure is 0.3 bar, and the steam is 0.912 dry. Draw a line
diagram of the plant and determine (a) the steam quality at the exhaust
from the H.P. turbine, (b) the power developed by the H.P. turbine, and
(c) the isentropic efficiency of the H.P. turbine.
(Ans. (a) 0.96, (b) 1125 kW, (c) 77%)
Solution:
Given steam flow rate
35
•
m = 21000 kg/h =
kg/s
6
1
HPT
4
2
3
5
LPT
6
BFP
From Steam Table at 17 bar 230°C
250°C
15 bar 200°C
h = 2796.8
2923.3
6.709
s = 6.455
∴
at 230°C
30
h = 2796.8 +
(2623.3 − 2796.8) = 2872.7 kJ/kg
50
30
(6.709 − 6.455) = 6.6074 kJ/kg
s = 6.455 +
50
20 bar
212.4°C
250°C
Page 210 of 265
Con
Vapour Power Cycles
Chapter 12
h = 2797.2
h = 2902.5
s = 6.3366
s = 6.545
∴
at 230°C
17.6
h = 2797.2 +
(2902.5 –2797.2) = 2846.4 kJ/kg
37.6
17.6
(6.545 − 6.3366) = 6.434 kJ/kg
s = 6.3366 +
37.6
∴
at 17 bar 230°C
2
h1 = 2872.7 + (2846.5 − 2872.7) = 2862.2 kJ/kg
5
2
s1 = 6.6074 + (6.434 − 6.6074) = 6.5381 kJ/kg
5
h2 = 871.8 + 0.957 × 1921.5 = 2710.7 kJ/kg ≈ h3
h4 = ?
∴
Mass flow through process heater
132.56
•
= 0.97597 kg/s = 3513.5 kg/h
= (m1 ) =
h1 − h 2
∴
Mass flow through HPT
= 17486.5 kJ/kg = 4.8574 kg/s
∴
21000 h5 = 17486.5 h4 + 3513.5 h3
... (i)
h6 = 289.3 = 0.912 × 2336.1 = 2419.8 kJ/kg
•
WT = m(h5 − h 6 )
WT
⎛ 1337.5 × 3600
⎞
+ h6 = ⎜
+ 2419.8 ⎟ = 2649.1 kJ/kg
21000
⎝
⎠
m
∴
h5 =
∴
From (i) h4 = 2636.7 kJ/kg
•
At 3–5 bar hg = 2731.6 kJ/kg so it is wet is quality x
(a)
∴
2636.7 = 584.3 + x × 2147.4 ⇒ x = 0.956
(b)
(c)
•
WHPT = m2 (h1 − h 4 )
17486.5
(2862.2 × 2636.7) kJ/kg = 1095 kW
=
3600
At 3.5 bar, sf – 1.7273, sfg = 5.2119 quality is isentropic
x = 0.923
6.5381 = 1.7273 + x × 5.2119
∴
h4s = 584.3 + 0.923 × 2147 .4 = 2566.4 kJ/kg
h − h4
2862.2 − 2636.7
∴ η isen. = 1
=
× 100% = 76.24%
2862.2 − 2566.4
h1 − h 4 s
Q.12.20
Solution:
In a cogeneration plant, the power load is 5.6 MW and the heating load is
1.163 MW. Steam is generated at 40 bar and 500°C and is expanded
isentropically through a turbine to a condenser at 0.06 bar. The heating
load is supplied by extracting steam from the turbine at 2 bar which
condensed in the process heater to saturated liquid at 2 bar and then
pumped back to the boiler. Compute (a) the steam generation capacity of
the boiler in tonnes/h, (b) the heat input to the boiler in MW, and (c) the
heat rejected to the condenser in MW.
(Ans. (a) 19.07 t/h, (b) 71.57 MW, and (c) 9.607 MW)
From steam table at 40 bar 500°C
Page 211 of 265
Vapour Power Cycles
Chapter 12
h1= 3445.3 kJ/kg
s1 = 7.090 kJ/kg
T
5
1
1 kg
7
6
4
Q0
2
m kg
(1 – m) kg
(1 – m )k g
3
S
→ at 2 bar
sf = 1.5301, sfg = 5.5967
∴
∴
7.090 = 1.5301 + x × 5.5967
x = 0.9934
h2 = 504.7 + 0.9934 × 2201.6
= 2691.8 kJ/kg
→ at 0.06 bar
sf = 0.521, sfg = 7.809
∴
∴
so
7090 = 0521 + x × 7.809 ⇒ x = 0.8412
h3 =151.5 + 0.8412 × 2415.9 = 2183.8 kJ/kg
h4 = 151.5 kJ/kg
h6 = 504.7 kJ/kg
WP4 − 5 = 0.001006 × (4000 – 6) = 4 kJ/kg
h5 = h4 + WP = 155.5 kJ/kg
WP6 − 7 = 0.001061 × (4000 – 100) = 4 kJ/kg
so
h7 = h6 + WP = 508.7kJ/kg
For heating load Qo = h2 – h6 = (2691.8 – 504.7) kJ/kg
= 2187.1 kJ/kg
For WT = (h1 – h2) + (1 – m) (h2 – h3)
= 753.5 + (1 – m) 508
= 1261.5 – 508 m
∴ Wnet = WT – WP4 −5 (1 − m) − m WP6 −7
= (1257.5 – 508 m) kJ/kg
If mass flow rate at ‘1’ of steam is w kg/s then
w (1257.5 – 508m) = 5600
wm × 2187.1 = 1163
From (i) & (ii) w = 4.668 kg/s = 16.805 Ton/h
∴
m = 0.11391 kg/kg of the generation
(a)
Steam generation capacity
boiler
= 16.805 t/h
Page of
212
of 265
...(i)
...(ii)
Vapour Power Cycles
Chapter 12
Q.12.21
Solution:
(b)
Heat input to the boiler
= W [(1 – m) (h1 – h5) + m (h1 – h7)] =15.169 MW
(c)
Heat rejection to the condenser
= (1 – m) (h3 – h4) = 8.406 MW
Steam is supplied to a pass-out turbine at 35 bar, 350°C and dry
saturated process steam is required at 3.5 bar. The low pressure stage
exhausts at 0.07 bar and the condition line may be assumed to be
straight (the condition line is the locus passing through the states of
steam leaving the various stages of the turbine). If the power required is
1 MW and the maximum process load is 1.4 kW, estimate the maximum
steam flow through the high and low pressure stages. Assume that the
steam just condenses in the process plant.
(Ans. 1.543 and 1.182 kg/s)
Form Steam Table 35 bar 350°C
6.743 + 6.582
= 6.6625 kJ/kg
s1 =
2
3115.3 + 3092.5
h1 =
= 3103.9 kJ/kg
2
sf = 1.7273
at 3.5 bar
sfg = 5.2119
∴
if condition of steam is x1
6.6625 = 1.7273 + x1 x 5.2119
x1 = 0.9469
∴ h2 = 584.3 + 0.9469 × 2147.4 = 2617.7 kJ/kg
At 0.07 bar
sf = 0.559
sfg = 7.717
∴
6.6625 = 0.559 + x × 7.717
⇒ x 2 = 0.7909
1 35 bar 330°C
T
6
4
3.5 bar
m kg
5
2
0.07 bar
(1 – m)
Q0
(1 – m) kg
3
S
∴
h3 = 163.4 + 07909 × 2409.1 = 2068.8 kJ/kg
h4 = 163.4 kJ/kg ∴ h6 = 584.3 kJ/kg
WP4 − 5 = 0.001007 (3500 –7) = 3.5 kJ/kg
Page 213 of 265
Vapour Power Cycles
Chapter 12
∴
h5 = h4 + WP4 − 5 = 166.9 kJ/kg
WP6 − 7 = 0.001079 (3500 – 350) = 3.4 kJ/kg
∴
h7 = h6 + WP6 − 7 = 587.7 kJ/kg
Let boiler steam generation rate = w kg/s
∴
WT = w [(h1 – h2) + (1 – m) (h2 – h3)]
Wnet = w [(h1 – h2) + (1 – m) (h2 – h3) – (1 – m) 3.5 – m × 3.4] kW
= w [486.2 + 545.4 – 542 m]
= w [1031.6 – 542 m] kW
Q = mw [h2 – h6] = mw (2033.4) kW
Here w [1031.6 – 542m] = 1000
… (i)
mw × 2033.4 = 1400
...(ii)
∴
w = 1.3311 kg/s
;
m = 0.51724 kg/kg of steam at H.P
∴
AT H.P flow 1.3311 kg/s
At L.P flow = (1 – m) w = 0.643 kg/s
Q.12.22
Solution :
Geothermal energy from a natural geyser can be obtained as a
continuous supply of steam 0.87 dry at 2 bar and at a flow rate of 2700
kg/h. This is utilized in a mixed-pressure cycle to augment the
superheated exhaust from a high pressure turbine of 83% internal
efficiency, which is supplied with 5500 kg/h of steam at 40 bar and 500 °c .
The mixing process is adiabatic and the mixture is expanded to a
condenser pressure of 0.10 bar in a low pressure turbine of 78% internal
efficiency. Determine the power output and the thermal efficiency of the
plant.
(Ans. 1745 kW, 35%)
From Steam Table 40 bar 500°C
h1 = 3445.3 kJ/kg
s1 = 7.090 kJ/kg – K
1
HPT η = 83%
2
5
3
1
Η = 78%
4
T
6
2s
4s
5
S
At 2 bar
hf = 504.7 kJ/kg, hfg = 2201.6 kJ/kg
Page 214 of 265
3
4
Vapour Power Cycles
Chapter 12
sf = 1.5301 kJ/kg, sfg = 5.5967 kJ/kg
7.090 = 1.4301 + x1 × 4.5967
∴
x1 = 0.99342
∴ h2s = 504.7 + 0.99342 × 2201.6 kJ/kg = 2691.8 kJ/kg
h − h2
ηisen. = 1
h1 − h 2s
∴
h2 – h1 – ηin (h1 – h2s)
= 3445.3 – 0.83 (3445.3 – 2691.8) = 2819.9 kJ/kg
s2 = 7.31 kJ/kg – K
From molier diagram
Adiabatic mixing
h5 = 504.7 + 0.87 × 2201.6 = 2420 kJ/kg
∴ h2 × 5500 + h5 × 2700 = h3 × (5500 + 2700)
∴ h3 = 2688.3 kJ/kg from molier dia at 2 bar 2688.3 kJ/kg
quality of steam x3
Then 504.7 + x 2 × 2201.6 = 2688.3 ⇒ x 3 = 0.9912
∴ s3 = 1.5301 + 0.9918 × 5.5967 = 7.081 kJ/kg – K
at 0.1 bar
sf = 0.649 + sfg = 7.501
∴ x4 × 7.501 + 0.649 = 7.081
⇒ x4 = 0.8575
∴
h4s = 191.8 + 0.8575 × 2392.8 = 2243.6 kJ/kg
5500
(h1 − h 2 ) = 955.47 kW
∴ WTH . P =
3600
8200
(h3 − h 4 s ) × 0.78 = 790 kW 790.08 kW
WTL . P =
3600
∴
WT = 1745.6 kW
5500
WP =
× 0.001010 × (4000 − 10) = 6.16 kW
3600
∴
Q.12.23
Wnet = 1739.44 kW
h5 = 191.8 kJ/kg, h6 = h5 + WP = 195.8 kJ/kg
∴ Heat input =
5500
(h1 − h 6 ) = 4964.5 kW
3600
∴
1739.44
× 100% = 35.04%
4964.5
η=
In a study for a space projects it is thought that the condensation of a
working fluid might be possible at - 40 °C . A binary cycle is proposed,
using Refrigerant 12 as the low temperature fluid, and water as the high
temperature fluid. Steam is generated at 80 bar, 500°C and expands in a
turbine of 81% isentropic efficiency to 0.06 bar, at which pressure it is
condensed by the generation of dry saturated refrigerant vapour at 30°C
from saturated liquid at -40°C. The isentropic efficiency of the R-12
turbine is 83%. Determine the mass ratio of R-12 to water and the
efficiency of the cycle. Neglect all losses.
(Ans. 10.86; 44.4%.)
Page 215 of 265
Vapour Power Cycles
Chapter 12
Solution :
at 80 bar 500°C
h1 = 3398.3 kJ/kg
s1 = 6.724 kJ/kg – K
80 bar
1
1 kg
0.06 bar
4
22
a
m
kg
b
T
d
c
S
at 0.06 bar if quality is x, then
6.724 = 0.521 + x 2 × 7.809
∴
x 2 = 0.79434
∴
h2s = 151.5 + 0.79434 × 2415.9 = 2070.5 kJ/kg
h3 = 151.5
WP = 0.001006 (8000 – 6) = 8 kJ/kg
∴
h4 = 159.5 kJ/kg
∴
WT = (h1 – h2s) × η = (3398.3 – 2070.5) × 0.81 = 1075.5 kJ/kg
∴
Wnet = WT – WP = 1067.5 kJ/kg
Q1 = h1 – h4 = 3238.8 kJ/kg
Q2 = h2 – h3 = h1 – η (h1 – h2s) – h3 = 2171.3 kJ/kg
For R-12
at 30°C saturated vapour
ha = 199.6 kJ/kg, p = 7.45 bar sg = 0.6854 kJ/kg – K
at 40°C
sfg = 0.7274, ∴ if dryness x b then
sf = 0,
xb × 0.7274 = 0.6854
∴
∴
hC = 0
⇒ x b = 0.94226
sf = 0, hfg = 169.0 kJ/kg
hb – 0.94226 × 169 = 159.24 kJ/kg
⎛ 0.66 + 0.77 ⎞
−3
WP = Vc (pa – pc ) = ⎜
⎟ × 10 (745 – 64.77) x = 0.4868 kJ/kg
2
⎝
⎠
∴
hb = hC + WP = 0.4868 kJ/kg
∴
Heat input = m (ha – hd)
= m (199.6 – 0.4868) = 199.11 m = 2171.3
∴
m = 10.905 kg of R-12/kg of water
Power output WTR = m (ha – hb) × η
= 10.905 × (199.6 – 159.24) × 0.83 = 365. 3 K
Page 216 of 265
Vapour Power Cycles
Chapter 12
∴
∴
Wnet = 364.8 kJ/kg of steam
Woutput = Wnet H2 O + Wnet R12
= (1067.5 + 364.8) = 1432.32 kJ/kg
Woutput
1432.32
η=
× 100% = 44.22%
=
3238.8
Heat input
∴
Q.12.24
Solution :
Steam is generated at 70 bar, 500°C and expands in a turbine to 30 bar
with an isentropic efficiency of 77%. At this condition it is mixed with
twice its mass of steam at 30 bar, 400°C. The mixture then expands with
an isentropic efficiency of80% to 0.06 bar. At a point in the expansion
where me pressure is 5 bar, steam is bled for feedwater heating in a
direct contact heater, which raises the feed water to the saturation
temperature of the bled steam. Calculate the mass of steam bled per kg
of high pressure steam and the cycle efficiency. Assume that the L.P.
expansion condition line is straight.
(Ans. 0.53 kg; 31.9%)
From Steam Table 70 bar 500°C
h1 = 3410.3 kJ/kg
s1 = 6.798 kJ/kg – K
s1 at 30 bar 400°C
h′3 = 3230.9 kJ/kg
s 3′ = 6.921 kJ/kg
From Molier diagram h2s = 3130 kJ/kg
∴
h2 = h1 – ηisentropic × (h1 – h2s)
= 3410.3 – 0.77 (3410.3 – 3130) = 3194.5 kJ/kg
For adiabatic mixing
1 × h2 + 2 × h′3 = 3 × h3
∴
∴
h3 = 3218.8 kJ/kg
s3 = 6.875 kJ/kg
(From Molier diagram)
h4′ = 2785 kJ/kg
h5s = 2140 kJ/kg
h5 = h3 – η(h5 – h5s) = 3218.8 – 0.80 (3218.8 – 2140) kJ/kg
1
70° bar 500°C
1kg
2
1m
2 kg
T
7
3 kg 6
8
3′ 3 30° bar 900°C
2s
5 bar
m
(3 – m)
(3 – m) 0.06 bar
5s
S
From S.L in H.P. h4 = 2920 kJ/kg
From Heat balance & heater
m × h4 + (3 – m)h7 = 3hg
h7 = h6 + WP
Page 217 of 265
Vapour Power Cycles
Chapter 12
= 151.5 + 0.001006 × (500 – 6) ≈ 0.5 kJ/kg
m × 2920 + (3 – m) × 152 = 3 × 640.1 = 152 kJ/kg
m = 0.529 kg
hg = 640.1
WTH . P = 1 × (h1 – h2) = (3410.3 – 3194.5) kJ/kg = 215.8 kJ/kg
∴
WTL . P = 3 (h3 – h4) + (3 – m) (h4 – h5)
= 3 (3218.8 – 2920) + (3 – 0.529) (2920 – 2355.8)
= 2290.5 kJ/kg of steam H.P
WP = (3 – m) (h7 – h6) + 2 × 0.001(3000 – 500)
+ 1 × 0.001 (7000 – 500) = 12.74 kJ/kg of H.P
∴ Wnet = (215.8 + 2290.5 – 12.74) kJ/kg & H.P steam
= 2493.6 kJ/kg of H.P steam
Heat input Q1 = (h1 – h10) + 2 (h3′ – h9)
∴
h10 + h8 + WP8 − 10 = 646.6 kJ/kg
h9 = h8 + WP8 − 9
= (3410.3 – 646.6) + 2 (3230.9 – 642.6)
= 7940.3 kJ/kg of H.P steam
∴
Q.12.25
Solution:
ηcycle =
= 642.6 kJ/kg
2493.6
× 100% = 31.4%
7940.3
An ideal steam power plant operates between 70 bar, 550°C and 0.075
bar. It has seven feed water heaters. Find the optimum pressure and
temperature at which each of the heaters operate.
Try please.
Q.12.26
In a reheat cycle steam at 550°C expands in an h.p. turbine till it is
saturated vapour. It is reheated at constant pressure to 400°C and then
expands in a I.p. turbine to 40°C. If the moisture content at turbine
exhaust is given to be 14.67%, find (a) the reheat pressure, (b) the
pressure of steam at inlet to the h.p. turbine, (c) the net work output per
kg, and (d) the cycle efficiency. Assume all processes to be ideal.
(Ans. (a) 20 bar, (b) 200 bar, (c) 1604 kJ/kg, (d) 43.8%)
Solution:
From S.T. at 40°C, 14.67% moisture
∴
x = 0.8533
1 p
1
3
T
6
5
p2
2
p3
4
S
Page 218 of 265
Vapour Power Cycles
Chapter 12
p3 = 0.0738 bar
∴
∴
(a)
hf = 167.6 kJ/kg; hfg = 2406.7 kJ/kg
h4 =167.6 + 0.85322 × 2406.7
= 2221.2 kJ/kg
sf = 0.573 kJ/kg – K
sfg = 7.685 kJ/kg – K
s4 = 0.573 + 0.8533 × 7.685 = 7.1306 kJ/kg
at 400°C and s4 = 7.1306 From Steam Table
Pr = 20 bar,
At 20 bar saturation
h2 = 2797.2 kJ/kg
∴
h3 = 3247.6 kJ/kg
S2 = 6.3366 kJ/kg – K
at 550°C and 6.3366 kJ/kg – k
(b)
From Steam Table
Pr = 200 bar
∴
h1 = 3393.5 kJ/kg
∴
h5 = 167.6 kJ/kg
WP = 0.001 × (20000 – 7.38) kJ/kg
∴
h6 = h5 + W = 187.6 kJ/kg
= 20 kJ/kg
∴
WT = (h1 – h2) + (h3 – h4) = 1622.7 kJ/kg
(c)
∴
Wnet = WT – WP = 1602.7 kJ/kg
Heat input Q1 = (h1 – h6) + (h3 – h2)
= (3393.5 – 187.6) + (3247.6 – 2797.2) kJ/kg
= 3656.3 kJ/kg
(d)
Q.12.27
Solution:
∴
η=
1602.7
× 100% = 43.83 %
3656.3
In a reheat steam cycle, the maximum steam temperature is limited to
500°C. The condenser pressure is 0.1 bar and the quality at turbine
exhaust is 0.8778. Had there been no reheat, the exhaust quality would
have been 0.7592. Assuming ideal processes, determine (a) reheat
pressure, (b) the boiler pressure, (c) the cycle efficiency, and (d) the
steam rate.
(Ans. (a) 30 bar, (b) 150 bar, (c) 50.51%, (d) l.9412 kg/kWh)
From 0.1 bar (saturated S.T.)
sf = 0.649 kJ/kg – K
sfg = 7.501 kJ/kg – K
∴
s4 = s3 = 0.649 + 0.8778 × 7.501 = 7.233 kJ/kg – K
s4 " = s1 = 0.649 + 0.7592 × 7.501 = 6.344 kJ/kg – K
h4 = 191.8 + 0.8778 × 2392.8 = 2292.2 kJ/kg
From super heated steam turbine at 500°C 7.233 kJ/kg
p3 = 30 bar
∴
h3 = 3456.5 kJ/kg, h2 = 2892.3 kJ/kg
From Molier diagram
Page 219 of 265
Vapour Power Cycles
Chapter 12
1
T
3
50° C
2
6
0.1 bar
5
4′
4
S
At 500°C and 6.344 kJ/kg – K
p1 = 150 bar,
h2 = 3308.6 kJ/kg – K
h5 = 191.8 kJ/kg,
WP = 0.001010 (15000 – 10) =15.14 kJ/kg
∴
h6 = 206.94 kJ/kg
∴
WT =(h1 – h2) + (h3 – h4) = 1580.6 kJ/kg
Wnet = WT – WP = 1565.46 kJ/kg
Q1 = (h1 – h6) + (h3 – h2) = 3665.86 kJ/kg – K
1565.45
∴
η=
≈ 42.7%
3665.86
Q.12.28
Solution:
In a cogeneration plant, steam enters the h.p. stage of a two-stage
turbine at 1 MPa, 200°C and leaves it at 0.3 MPa. At this point some of
the steam is bled off and passed through a heat exchanger which it
leaves as saturated liquid at 0.3 MPa. The remaining steam expands in
the I.p. Stage of the turbine to 40 kPa. The turbine is required to
produce a total power of 1 MW and the heat exchanger to provide a
heating rate of 500 kW. Calculate the required mass flow rate of steam
into the h.p. stage of the turbine. Assume (a) steady condition
throughout the plant, (b) velocity and gravity terms to be negligible, (c)
both turbine stages are adiabatic with isentropic efficiencies of 0.80.
(Ans. 2.457 kg/s)
From S.T at 1 MPa 200°C
i.e
10 bar 200°C
h1 = 2827.9 kJ/kg
s1 = 6.694 kJ/kg-K
At 3 bar
sf = 1.6716 sfg = 5.3193
∴
∴
∴
∴
6.694 = 1.6716 + x′ × 5.3193
x′ = 0.9442
h2s = 561.4 + 0.9242 × 2163 = 2603.9 kJ/kg
h2 = h1 – η(h1 – h2s) = 2827.9 – 0.8 (2827.9 – 2603.0) kJ/kg
= 2648.7 kJ/kg
Page 220 of 265
Vapour Power Cycles
Chapter 12
1 kg
1
T
5
(1– m)kg
4
m kg
6
2s
Q0
(1– m)kg
2
(1–m)kg
3s
3
S
This is also wet so
s2 = 1.6716 + x2′ × 5.319.3 [2648.7 = 561.4 + x2 × 2163.2]
= 6.8042 kJ/kg – K
If at 3s condition of steam is x3 then 40 kPa = 0.4 bar
⇒ x 3 = .8697
6.8042 = 1.0261 + x 3 × 6.6440
∴
∴
h3s = 317.7 + 0.8697 × 2319.2 = 2334.4 kJ/kg
h3 = h2 – η (h2 – h3s) = 2397.3 kJ/kg
h4 = 317.7 kJ/kg
WP4 s = 0.001 × (1000 – 40) ≈ 1 kJ/kg
∴
h5 = 318.7 kJ/kg
h6 = 562.1 kJ/kg, WP6 − 7 = 0.001 × (1000 – 300)
= 0 0.7 kJ/kg
h7 = 562.1 kJ/kg
∴
∴
WT = (h1 – h2) + (1 – m) (h2 –h3)
Wnet= (h1 – h2) + (1 – m) (h2 – h3) – (1 – m) 1 – m × 0.7
= (429.6 – 251.1 m) kJ/kg of steam of H.P
∴
Process Heat = m × (h2 – h6) = m × 2087.3
If mass flow rate of w then
w (429.6 – 251.1 m) = 1000
mw × 2087.3 = 500
w = 2.4678 kg/s
∴
required mass flow rate in H.P = 2.4678 kg/s
∴
Page 221 of 265
Page 222 of 265
Gas Power Cycles
Chapter 13
13. Gas Power Cycles
Some Important Notes
1. Compression ratio, (rc)
rc =
Volume at the begining of compression (V1 )
Volume at the end of compression (V2 )
∴
rc =
2
p
V1
bigger term
=
V2
smaller term
1
V
2. Expansion ratio,( re)
1
Volume at the end of expansion (V2 )
re =
Volume at the begining of expansion (V1 )
p
2
V
bigger term
∴ re = 2 =
V1
smaller term
3. Cut-off ratio, ρ =
V
volume after heat addition (v 2 )
volume before heat addition (v1 )
(For constant Pressure heating)
∴
ρ=
V2
V1
bigger term
=
smaller term
Relation
p
1
rc = re . ρ
2
V
4. Constant volume pressure ratio,
∝=
Q
Pressure after heat addition
Pressure before heat addition
[For constant volume heating)
Page 223 of 265
Gas Power Cycles
Chapter 13
2
p
bigger term
2
α = p = smaller term
1
∴
Q
p
1
V
5. Pressure ratio, rP
Pressure after compresion or before expansion ⎛ p2 ⎞
Pressure before compresion or after expansion ⎜⎝ p1 ⎟⎠
=
p
p2
∴ rP = p
1
6.
Q1
2
3
pV = C
1
Q2
4
V
Carnot cycle: The large back work (i.e compressor work) is a big draw back for the
Carnot gas cycle, as in the case of the Carnot Vapour cycle.
7.
Stirling Cycle: comparable with Otto.
8.
Ericsson Cycle: comparable with Brayton cycle.
9.
The regenerative, stirling and Ericsson cycles have the same efficiency as the
carnot cycle, but much less back work.
10.
Air standards cycles
a.
Otto cycle (1876)
η =1−
1
γ −1
c
r
V=
3
p
T
4
2
1
2
3
V=C
4
1
V
S
1
For Wmax;
C
2( γ − 1)
T
rc = ⎛⎜ max ⎞⎟
⎝ Tmin ⎠
Page 224 of 265
Gas Power Cycles
Chapter 13
Diesel cycle (1892)
p
V
=
C
b.
2
4
T
V=C
2
4
1
1
V
C
3
3
p
=
S
(ρ γ − 1)
η = 1 − γ −1
rc . γ(ρ − 1)
C. Dual or Limited pressure or mixed cycle
3
p
4
2
T
5
1
Where α =
V=C
5
1
V
η=1−
4 p=C
3
V=C
2
S
γ
γ −1
c
r
( αρ − 1)
[( α − 1) + αγ(ρ − 1)]
p3
p2
Comparison of Otto, Diesel and Dual cycle
a. With same compression ratio and heat rejection
4
∴
ηotto > ηDual > ηDiesel
p
3
Q1
5
6
2
7
1
V
Page 225 of 265
Q2
Gas Power Cycles
Chapter 13
b.
For the Same maximum Pressure and Temperature (also heat rejection same)
Q1
5
4
p=C
6
6
V=C
3
p
5
T
2
V=C
4
7
Q2
3
2
1
V
V=C
7
1
ηDiesel > ηDual > ηOtto
11.
Brayton cycle
η= 1−
1
γ −1
c
r
1
=1−
γ −1
γ
P
r
p=C
3
2
T
p=C
4
1
S
∴
Brayton cycle efficiency depends on either compression ratio ( rc ) or Pressure ratio
rp
* For same compression ratio
⎡⎣ηOtto = ηBrayton ⎤⎦
γ
⎛ T ⎞ ( γ − 1)
a. For Maximum efficiency ( rp ) max = ⎜ max ⎟
⎝ Tmin ⎠
ηmax = ηCarnot
∴
b.
=1−
For Maximum work
γ
(i)
⎛ T ⎞ 2( γ − 1)
( rp ) opt. = ⎜ max ⎟
⎝ Tmin ⎠
Page 226 of 265
Tmin
Tmax
Gas Power Cycles
Chapter 13
2
Tmin
and Wnet, max = Cp [ Tmax − Tmin ]
Tmax
∴
ηcycle = 1 −
(ii)
If isentropic efficiency of Turbine is η T and compressor is ηc then
γ
T ⎞ 2( γ − 1)
⎛
( rp ) opt. = ⎜ ηT ηC max ⎟
Tmin ⎠
⎝
Question and Solution (P K Nag)
In a Stirling cycle the volume varies between 0.03 and 0.06 m3 , the
maximum pressure is 0.2 MPa, and the temperature varies between
540°C and 270°C. The working fluid is air (an ideal gas). (a) Find the
efficiency and the work done per cycle for the simple cycle. (b) Find the
efficiency and the work done per cycle for the cycle with an ideal
regenerator, and compare with the Carnot cycle having the same
isothermal heat supply process and the same temperature range.
(Ans. (a) 27.7%, 53.7 kJ/kg, (b) 32.2%)
Q13.1
Solution:
Given V1 = 0.06 m3 = V4
V2 = 0.03 m3 = V3
p3 = 200 kPa
3
Q1
T1 = T2 = 270°C = 543 K
T3 = T4 = 540°C = 813 K
p
2
p3 V3
200 × 0.03
=
= 0.025715 kg
R T3
0.287 × 813
∴ Q1 = 0.025715 × 0.718 (813 – 543) kJ = 4.985 kJ
Here m =
⎛V ⎞
pdV = m R T3 ln ⎜ 4 ⎟
3
⎝ V3 ⎠
pV = mRT = C
3
⎛V ⎞
W1 – 2 = ∫ pdV = m RT1 ln ⎜ 1 ⎟
1
⎝ V2 ⎠
∴p=
∫
4
T=C
1
∴ Heat addition Q1 = Q2 – 3 = m c v (T3 – T2)
W3 – 4 =
T=C
4
mRT
V
⎛V ⎞
m(RT3 − RT1 ) ln ⎜ 1 ⎟
⎝ V2 ⎠ =
∴η=
4.985
Page 227 of 265
Q2
Gas Power Cycles
Chapter 13
0.025715 × 0.287 (813 − 543) ln 2
×100%= 27.7%
4.985
Work done = 1.3812 kJ = 53.71 kJ/kg
For ideal regeneration
543
η= 1−
= 33.21%
813
Q13.2
An Ericsson cycle operating with an ideal regenerator works between
1100 K and 288 K. The pressure at the beginning of isothermal
compression is 1.013 bar. Determine (a) the compressor and turbine work
per kg of air, and (b) the cycle efficiency.
(Ans. (a) wT = 465 kJ/kg, wC = 121.8 kJ/kg (b) 0.738)
Solution:
Given T1 = T2 = 288 K
T3 = T4 = 1100 K
p1 = 1.013 bar = 101.3 kPa
RT1
= 0.81595 m3/kg
∴
V1 =
p1
Q1
2
p
S
3
T=C
1
T=C
Q2
4
V
⎛V ⎞
WC = RT1 ln ⎜ 1 ⎟
⎝ V2 ⎠
⎛V ⎞
WT = R T3 ln ⎜ 4 ⎟
⎝ V3 ⎠
p3 = p 2 ; p1 = p 4
∴η= 1−
288
= 73.82%
1100
∴ W = η Q1 = η CP (T1 –T2)
∴
p1 V1 p2 V2
=
T1
T2
∴
V1
T ⎛p ⎞ ⎛p ⎞
= 1 × ⎜ 2 ⎟= ⎜ 2 ⎟
V2
T2 ⎝ p1 ⎠ ⎝ p1 ⎠
= 0.7382 × 1.005 (1100 – 288) kJ/kg = 602.4 kJ/kg
∴ Q2 – 3 = CP (T3–T2) = Cv (T3 – T2) +
Q13.5
p 2 (V3 – V2)
An engine equipped with a cylinder having a bore of 15 cm and a stroke
of 45 cm operates on an Otto cycle. If the clearance volume is 2000 cm3 ,
compute the air standard efficiency.
(Ans.47.4%)
Page 228 of 265
Gas Power Cycles
Chapter 13
Solution:
V2 = 2000 cm3 = 0.002 m3
V1 = V2 + S.V.
= 0.002 +
3
π × 0.152
× 0.45 = 0.009952 m3
4
p
V1
= 4.9761
V2
1
ηair std = 1 − γ − 1 = 47.4%
rc
∴ rc =
Q13.10
4
2
1
V1
V2
SV
VCL
Two engines are to operate on Otto and Diesel cycles with the following
data: Maximum temperature 1400 K, exhaust temperature 700 K. State of
air at the beginning of compression 0.1 MPa, 300 K.
Estimate the compression ratios, the maximum pressures, efficiencies,
and rate of work outputs (for 1 kg/min of air) of the respective cycles.
(Ans. Otto-- rk = 5.656, p max = 2.64 MPa, W = 2872 kJ/kg, η = 50%
Diesel- rk , = 7.456, p max = 1.665 MPa, W = 446.45 kJ/kg, η = 60.8%)
T3 = 1400 K
T4 = 700 K
p1 = 100 kPa
Solution:
∴
T1 = 300 K
RT1
= 0.861 m3/kg
v1 =
p1
∴
⎛p ⎞
T3
= ⎜ 3⎟
T4
⎝ p4 ⎠
γ −1
γ
⎛v ⎞
= ⎜ 4⎟
⎝ v3 ⎠
γ −1
V=C
3
3
V=C
T
Q1
p
4
2
2
4
Q2
1
1
S
∴
∴
⎛ 1400 ⎞ ⎛ v1 ⎞
⎜ 700 ⎟ = ⎜ v ⎟
⎝
⎠ ⎝ 2⎠
v2 =
v1
1
γ −1
=
γ −1
0.861
1
2 0.9
2
= 0.1522 m3/kg
⎛v ⎞
T
∴ 2 = ⎜ 1⎟
T1
⎝ v2 ⎠
γ −1
= (5.657)0.4 × 300 = 600 K
Page 229 of 265
Gas Power Cycles
Chapter 13
1
∴
p3
p
= 2
T3
T2
∴
W
∴
η=
Diesel
γ
⎛v ⎞
p2
= ⎜ 1 ⎟ ⇒ P2 = 1131.5 kPa
p1
⎝ v2 ⎠
T
1400
⇒ p3 = 3 × p2 =
× 1131.5 kPa = 2.64 MPa
600
T2
v
rc = 1 = 2 γ − 1 = 5.657
v2
= Q1 – Q2 = Cv (T3 – T2) – Cv (T4 – T1)
= 0.718 [(1400 – 600) – (700 – 300)] kJ/kg = 287.2 kJ/kg.
Q1 − Q2
287.2
= 0.5 ≈ 50%
=
0.718 (1400 − 600)
Q1
T3 = 1400 K
T4 = 700 K
T1 = 300 K
∴ v1 = 0.861 m3/kg
p1 = 100 kPa
T3 ⎛ v 4 ⎞
= ⎜ ⎟
T4 ⎝ v 3 ⎠
∴
γ −1
0.4
1400 ⎛ v1 ⎞
∴
= ⎜ ⎟
700
⎝ v2 ⎠
1
v
∴ 1 = 2 0.4 = 22.5
v3
v1
∴ v3 = 3.5
= 0.1522 m3/kg
2
RT3
0.287 × 1400
= 2639.9 kPa
p3 =
=
0.1522
V3
2
3
p
4
1
∴
p 2 = p3
∴
T2 ⎛ p2 ⎞
= ⎜ ⎟
T1 ⎝ p1 ⎠
V
γ −1
γ
⎛v ⎞
= ⎜ 1⎟
⎝ v2 ⎠
γ −1
1
∴ T2 = 764 K
1
⎛ p ⎞γ
v
2639.9 ⎞1.4
rc = 1 = ⎜ 2 ⎟ = ⎛⎜
⎟ = 10.36
v 2 ⎝ p1 ⎠
⎝ 100 ⎠
p1 = p3 = 2.64 MPa
Q1 = Q2 – 3 = CP (T3 – T2) = 1.005 (1400 – 764) kJ/kg
= 638.84 kJ/kg
Page 230 of 265
Gas Power Cycles
Chapter 13
Q2 = Q4 – 1 = Cv (T4 –T1) = 0.718 (700 – 300) = 287.2 kJ/kg
W = Q1 – Q2 = 351.64 kJ/kg
∴
η=
Q13.11
W
351.64
=
= 55%
638.84
Q1
An air standard limited pressure cycle has a compression ratio of 15 and
compression begins at 0.1 MPa, 40°C. The maximum pressure is limited
to 6 MPa and the heat added is 1.675 MJ/kg. Compute (a) the heat
supplied at constant volume per kg of air, (b) the heat supplied at
constant pressure per kg of air, (c) the work done per kg of air, (d) the
cycle efficiency, (e) the temperature at the end of the constant volume
heating process, (f) the cut-off ratio, and (g) the m.e.p. of the cycle.
(Ans. (a) 235 kJ/kg, (b) 1440 kJ/kg, (c) 1014 kJ/kg,
(d) 60.5%, (e) 1252 K, (f) 2.144 (g) 1.21 MPa)
Solution:
rc =
v1
= 15
v2
p1 = 100 kPa
∴ v1 =
T1 = 40°C = 313 K
3
p
2
RT1
= 0.89831 m3/kg
p1
4
Q34
Q23
5
1
V
p3 = p 4 = 6000 kPa
Q2 – 4 = 1675 kJ/kg
∴
⎛v ⎞
T2
= ⎜ 1⎟
T1
⎝ v2 ⎠
γ− 1
= (15)1.4 – 1 ⇒ T2 = 924.7 K
γ
⎛v ⎞
p2
= ⎜ 1 ⎟ = 151.4 ⇒ p 2 = 4431 kPa
p1
⎝ v2 ⎠
∴
∴
∴
p V
p
p2 V2
6000
× 924.7 = 1252 K
= 3 3 ⇒ T3 = 3 × T2 =
p2
4431
T2
T3
Q2 – 4 = Cv (T3 – T2) + CP (T4 – T3) = 1675
T4 = T3 + 1432.8 k = 2684.8 K
Page 231 of 265
Gas Power Cycles
Chapter 13
∴
RT4
= 0.12842 m3/kg.
p4
∴
v4 =
∴
⎛v ⎞
T4
= ⎜ 5⎟
T5
⎝ v4 ⎠
γ −1
⎛v ⎞
= ⎜ 1⎟
⎝ v4 ⎠
γ −1
⇒
T4
= 2.1773 ∴ T5 = 1233 K
T5
(a) Heat supplied at constant volume = Cv (T3 – T2) = 235 kJ/kg
(b) Heat supplied at constant Pressure = (1675 – 235) = 1440 kJ/kg
(c) Work done = Q1 – Q2 = 1675 – Cv (T5 – T1) = 1014.44 kJ/kg
Q1 − Q2
1014.44
× 100% = 60. 56%
=
1675
Q1
(e) Temperature at the end of the heating (T3) = 1252 K
(d) Efficiency η =
(f) Cut-off ratio (ρ) =
v4
0.12842
=
= 2.1444
0.05988
v3
[∴ v3 =
(g) m. e. p.
∴ pm
(V1 –
∴ pm =
Q13.13
V2 ) = W
RT3
= 0.059887]
p3
1014.44
= 1209.9 kPa = 1.2099 MPa
v
v1 − 1
15
Show that the air standard efficiency for a cycle comprising two
constant pressure processes and two isothermal processes (all
reversible) is given by
η=
(T1 − T2 ) ln ( rp )
(γ −1) / γ
(γ −1) / γ
T1 ⎡1 + ln ( rp )
− T2 ⎤
⎢⎣
⎥⎦
Where T1 and T2 are the maximum and minimum temperatures of the
cycle, and
rp is the pressure ratio.
Page 232 of 265
Gas Power Cycles
Chapter 13
Solution:
1
T
4
2
Q1
4
Q1
1
p
Q2
T=C
T=C
3
3
Q2
V
S
2 dV
p
p
V
∴ W1 – 2 = ∫ pdV = RT1 ∫
rP = 4 = 1
= RT1 ln 2 = RT1 ln rP
1 V
p3
p2
V1
Q1 – 2 = 0 + W1 – 2
4
⎛V ⎞
⎛V ⎞
W3 – 4 = ∫ pdV = RT3 ln ⎜ 4 ⎟ = − RT3 ln ⎜ 3 ⎟ = – RT3 ln rP .
3
⎝ V4 ⎠
⎝ V3 ⎠
⎛V ⎞
∴
Wnet = W1 – 2 + W3 – 4 = R(T1 − T3 ) ln ⎜ 2 ⎟
⎝ V1 ⎠
= R (T1 – T2) ln rP .
Constant pressure heat addition = CP (T1 – T4)
γR
T1 = Tmax
(T1 − T4 )
=
γ−2
γR
(T1 − T2 )
=
T2 = Tmin.
γ −1
Total heat addition (Q1)
η =
Multiply
= Q12 + const Pr.
γR
(T1 − T2 )
= RT1 ln rP +
γ −1
R (T1 − T2 ) ln rP
⎡
⎤
γ
R ⎢(T1 ln rP +
(T1 − T2 ) ⎥
γ −1
⎣
⎦
γ −1 γ γ
,D ,N
γ
⎛ γ − 1) ⎞
⎜ γ ⎟ (T1 − T2 ) ln rp
⎠
= ⎝
γ −1
(T1 − T2 ) +
T1 ln rp
γ
η=
⎛ γ −1⎞
⎜
⎟
γ ⎠
(T1 − T2 ) ln rp⎝
⎛ γ − 1⎞
⎜
⎟
⎝ γ ⎠
p
T1 [1 + ln r
] − T2
Page 233 of 265
2
Gas Power Cycles
Chapter 13
Q13.14
Obtain an expression for the specific work done by an engine working
on the Otto cycle in terms of the maximum and minimum
r
Temperatures of the cycle, the compression ratio k , and constants of the
working fluid (assumed to be an ideal gas).
Hence show that the compression ratio for maximum specific work
output is given by
1 / 2(1−γ )
⎛T ⎞
rk = ⎜ min ⎟
⎝ Tmax ⎠
Solution:
Tmin = T1
Tmax = T3
Q1 = Cv (T3 – T2)
Q2 = Cv (T4 – T1)
∴
W = Q1 – Q2
= Cv [(T3 – T2) – (T4 – T1)]
⎛v ⎞
T
Hence 2 = ⎜ 1 ⎟
T1
⎝ v2 ⎠
γ −1
∴ T2 = T1 rc
⎛v ⎞
T
And 4 = ⎜ 3 ⎟
T3
⎝ v4 ⎠
=x
γ −1
∴ T4 = T3 . rc−( r − 1)
Then
γ −1
⎛v ⎞
= ⎜ 2⎟
⎝ v1 ⎠
= rcγ − 1
γ −1
= rc−( γ − 1) Let rcγ − 1
3
T
= 3
x
p
T
⎡
⎤
W = Cv ⎢ T3 − T1x − 3 + T1 ⎥
x
⎣
⎦
dW
For maximum W,
=0
dx
T
⎡
⎤
∴ Cv ⎢0 − T1 + 32 + 0 ⎥ = 0
x
⎣
⎦
T
∴ x2 = 3
T1
∴ rcγ − 1 =
T3
=
T1
Q1
4
Q2
2
1
Tmax
Tmin
1
1
⎛ T ⎞ 2(1 − γ )
⎛ T ⎞ 2( γ − 1)
Proved.
= ⎜ min ⎟
∴ rc = ⎜ max ⎟
⎝ Tmin ⎠
⎝ Tmax ⎠
Q13.15
A dual combustion cycle operates with a volumetric compression ratio rk
= 12, and with a cut-off ratio 1.615. The maximum pressure is given by
pmax = 54 p1 ' where p1 is the pressure before compression. Assuming
Page 234 of 265
Gas Power Cycles
Chapter 13
indices of compression and expansion of 1.35, show that the m.e.p. of the
cycle
pm = 10 p1
Hence evaluate (a) temperatures at cardinal points with T1 = 335 K, and (b
Cycle efficiency.
(Ans. (a) T2 = 805 K, p2 = 29.2 p1 ' T3 = 1490 K,
T4 = 2410 K, T5 = 1200 K, (b) η = 0.67)
Solution:
Here
v1
= rc = 12
v2
v4
= ρ = 1.615
v3
pv1.35 = C, n = 1.35
p max = p3 = p 4 = 54 p1
4
3
3
T
2
4
p
5
5
2
1
1
V
S
∴
And
⎛v ⎞
T2
= ⎜ 1⎟
T1
⎝ v2 ⎠
⎛v ⎞
p2
= ⎜ 1⎟
p1
⎝ v2 ⎠
p
p2
= 3
T3
T2
n −1
∴ T2 = T1 × (12 )
(1.35
– 1)
= 2.3862 T1
n
∴ p2 = p1 × (12)1.35 = 28.635 p1
∴ T3 =
p3
54p1
× T2 =
× 2.3862 T1 = 4.5 T1
T2
28.635p1
⎛v ⎞
v3 = v 2 = ⎜ 1 ⎟
⎝ 12 ⎠
∴
∴
1.615
v1 = 0.13458 v1
12
Pv
p4 v 4
p3 = p 4
= 3 3
T3
T4
v
T4 = T3 × 4 = 1.615 T3 = 1.615 × 4.5 T1 = 7.2675 T1
v3
v 4 = ρ v3 =
n −1
∴
∴
∴
n −1
⎛v ⎞
⎛v ⎞
T5
= ⎜ 4⎟
= ⎜ 4⎟
T4
⎝ v1 ⎠
⎝ v5 ⎠
T5 = 3.6019 T1
W = [Cv (T3 – T2) + CP (T4 – T3) – Cv (T5 – T1) = 2.4308 T1 kJ/kg.
Page 235 of 265
Gas Power Cycles
Chapter 13
p m (v1 – v2) = W
2.4308 p1
2.4308 T1
= 9.25 p1
=
v
11
×R
v1 − 1
12
12
2.4308 T1
× 100 % = 56.54%
(b) ∴ η =
4.299 T1
(a)
T1 = 335 K, T2 = 799.4 K, T3 = 1507.5 K, T4 = 2434.6 K,
T5 = 1206.6 K.
pm =
∴
Q13.16
Recalculate (a) the temperatures at the cardinal points, (b) the m.e.p.,
and (c) the cycle efficiency when the cycle of Problem 13.15 is a Diesel
cycle with the same compression ratio and a cut-off ratio such as to give
an expansion curve coincident with the lower part of that of the dual
cycle of Problem 13.15.
(Ans. (a) T2 = 805 K, T3 = 1970 K, T4 = 1142 K
(b) 6.82 p1 , (c) η = 0.513)
Solution:
v
Given 1 = 12 = rc
v2
v3
= ρ = 1.615
v2
∴ T3 =
Then
2
v3
× T2 = 1.615 × 799.4 = 1291 K
v2
⎛v ⎞
T2
= ⎜ 1⎟
T1
⎝ v2 ⎠
3
p
4
n −1
∴ T2 = T1 (12 )
1.35– 1
1
1=
V
799.4 K
n
⎛v ⎞
p
But 2 = ⎜ 1 ⎟
p1
⎝ v2 ⎠
Continue to try…..
Q13.19
Solution:
In a gas turbine plant working on the Brayton cycle the air at the inlet is
at 27°C, 0.1 MPa. The pressure ratio is 6.25 and the maximum
temperature is 800°C. The turbi- ne and compressor efficiencies are each
80%. Find (a) the compressor work per kg of air, (b) the turbine work per
kg of air, (c) the heat supplied per kg of air, (d) the cycle efficiency, and
(e) the turbine exhaust temperature.
(Ans. (a) 259.4 kJ/kg, (b) 351.68 kJ/kg, (c) 569.43 kJ/kg,
(d) 16.2%, (e) 723 K)
Maximum Temperature
T1 = 800° C = 1073 K
p3 = 100 kPa
T3 = 300 K
Page 236 of 265
Gas Power Cycles
Chapter 13
rP = 6.25
p4
= 6.25
p3
1
4s
4
p
4s
T
1
4
2
2s
V
S
p 4 = 625 kPa
∴
p 2 = 100 kPa
p1 = p 4
∴
v3 =
2s
3
3
RT3
= 0.861
p3
v3 = 0.861
1
γ
⎛ p ⎞γ
p4 ⎛ v 3 ⎞
v
= ⎜ ⎟ ∴ 4 = ⎜ 3⎟
p3 ⎝ v 4 ⎠
v3
⎝ p4 ⎠
T3 = 300 K
1
⎛ p ⎞4
v 4 = v3 × ⎜ 3 ⎟
⎝ p4 ⎠
p 2 = p3
γ −1
⎛v ⎞
T4
= ⎜ 3⎟
T3
⎝ v4 ⎠
p 4 = 625 kPa
∴ T4 = T3 × (3.70243)0.4
T4 s − T3
T4 − T3
T4s = 506.4 K
∴ 0.8 =
⎛p ⎞
T1
= ⎜ 1⎟
T2s
⎝ p2 ⎠
γ −1
γ
T4 = 558 K
v4s
= 0.23255
∴ T4 = 558
T2s = 635.6 K
⎛p ⎞
= ⎜ 4⎟
⎝ p3 ⎠
γ −1
γ
= 1.68808
T2 = 723 K
T1 − T2
⇒ T1 – T2 = 350
T1 − T2s
T2 = T1 – 350 = 723 K
η=
∴
(a) Compressor work (Wc) = (h4 – h3) = Cp(T4 – T3) = 259.3 kJ/kg
Page 237 of 265
Gas Power Cycles
Chapter 13
(b) Turbine work ( WT ) = (h1 – h2) = Cp(T1 – T2) = 351.75 kJ/kg
(c) Heat supplied (Q1) = Cp(T1– T4) = 517.6 kJ/kg
WT − WC
= 17.86%
Q1
(d) Cycle efficiency (η) =
(e) Turbine exhaust temperature (T2) = 723 K
Q13.27
A simple gas turbine plant operating on the Brayton cycle has air inlet
temperature 27°C, pressure ratio 9, and maximum cycle temperature
727°C. What will be the improvement in cycle efficiency and output if the
turbine process Is divided into two stages each of pressure ratio 3, with
intermediate reheating to 727°C?
(Ans. - 18.3%, 30.6%)
Solution:
p2
3
1000 K
p1
T
562 K
2
4
300 K
1
(a)
For (a)
S
T1 = 300 K
p2
=9
p1
T3 = 1000 K
⎛p ⎞
T2 = ⎜ 2 ⎟
⎝ p1 ⎠
∴
⎛p ⎞
T4
= ⎜ 4⎟
T3
⎝ p3 ⎠
533.8 K
γ− 1
γ
⎛p ⎞
= ⎜ 1⎟
⎝ p2 ⎠
γ −1
γ
γ −1
γ
× T1 = 562 k
⎛1 ⎞
= ⎜ ⎟
⎝9⎠
γ −1
γ
∴ T4 =
T3
9
Page 238 of 265
γ −1
γ
= 533.8 K
Gas Power Cycles
Chapter 13
p2
pi
1000 K
1000K 3
T
562 K 2
4
1
5
730.6 K
p1
6730.6 K
300 K
S
(b)
⎛p ⎞
T4
= ⎜ i⎟
For (b)
T3
⎝ p2 ⎠
⎛p ⎞
T6
= ⎜ 1⎟
T5
⎝ pi ⎠
∴ For (a)
∴
For (b)
γ −1
γ
γ− 1
γ
⎛1 ⎞
∴ T4 = T3 × ⎜ ⎟
⎝3⎠
⎛1 ⎞
∴ T6 = T5 × ⎜ ⎟
⎝3⎠
γ −1
γ
γ− 1
γ
= 730.6 K
= 730.6 K
W = (h3 – h4) – (h2 – h1)
= Cp [T3 – T4) – (T2 – T1)] = 205.22 kJ/kg
Q = h3 – h2 = CP (T3 – T2) = 440.19 kJ/kg
η = 46.62 %
W = (h3 – h4) + (h5 – h6) – (h2 – h1)
= CP [(T3 – T4) + (T5 – T6) – (T2 – T1)] = 278.18 kJ/kg
Q = h3 – h2 + h5 – h4 = CP [(T3 – T2) + (T5 – T4)] = 710.94 kJ/kg
∴
η = 39.13 %
∴ Efficiency change =
39.13 − 46.62
× 100% = –16.07 %
46.62
Work output change =
278.18 − 205.22
× 100 = 35.6%
205.22
Q13.28
Obtain an expression for the specific work output of a gas turbine unit
in terms of pressure ratio, isentropic efficiencies of the compressor and
turbine, and the maximum and minimum temperatures, T3 and T1 •
Hence show that the pressure ratio
⎛
T ⎞
rp = ⎜ηT ηC 3 ⎟
T1 ⎠
⎝
rp
for maximum power is given by
γ / 2(γ −1)
Page 239 of 265
Gas Power Cycles
Chapter 13
If T3 = 1073 K, T1 = 300 K, ηC = 0.8, ηT = 0.8 and γ = 1.4 compute the
optimum
Value of pressure ratio, the maximum net work output per kg of air, and
corresponding cycle efficiency.
(Ans. 4.263, 100 kJ/kg, 17.2%)
Solution:
T1 = Tmin
T3 = Tmax
p2
3
Hence
⎛ p2 ⎞
⎟
⎝ p1 ⎠
T2s = T1 × ⎜
γ− 1
γ
T
2s
p1
2
4
4s
γ− 1
= T1 × rP γ
1
S
Let
∴
∴
rp
γ− 1
γ
=x
T2s = x T1
If isentropic efficiency and compressor is ηc
ηc =
∴
T2s − T1
T2 − T1
T2 = T1 +
⎡
T2s − T1
x − 1⎤
= T1 ⎢1 +
⎥
ηC ⎦
ηC
⎣
⎛p ⎞
Similarly T4s = T3 ⎜ 4 ⎟
⎝ p3 ⎠
∴
γ −1
γ
⎛p ⎞
= T3 ⎜ 1 ⎟
⎝ p2 ⎠
γ− 1
γ
=
If isentropic efficiency of turbine is ηT
Then ηT =
T3 − T4
⇒ – T3 + T4 = ηT (T4s – T3)
T3 − T4S
⎛T
⎞
T4 = T3 + ηT ⎜ 3 − T3 ⎟
⎝ x
⎠
⎡
⎛1
⎞⎤
= T3 ⎢1 + ηT ⎜ − 1 ⎟ ⎥
⎝x
⎠⎦
⎣
Page 240 of 265
T3
x
Gas Power Cycles
Chapter 13
∴
Specific work output
W = (h3 – h4) – (h2 – h1)
= CP [(T3 – T4) – (T2 – T1)]
⎡ ⎛
T ⎞ xT − T1 ⎤
= CP ⎢ηT ⎜ T3 − 3 ⎟ − 1
⎥ kJ/kg
x ⎠
ηC ⎦
⎣ ⎝
⎡
⎛
⎞
⎢
⎜
1 ⎟ Tmin
= CP ⎢ηT Tmax ⎜1 − γ− 1 ⎟ −
ηC
⎢
⎜
rp γ ⎟⎠
⎝
⎣
For maximum Sp. Work
⎤
⎛ γ−γ 1
⎞⎥
⎜ rp − 1 ⎟ ⎥ kJ/ kg
⎜
⎟
⎝
⎠⎥
⎦
dW
=0
dx
⎡η T
T ⎤
dW
= CP ⎢ T 2 3 − 1 ⎥ = 0
dx
ηC ⎦
⎣ x
∴
∴
x2 = ηT ηC
∴
x=
T3
TT1
ηT ηC
Tmax
Tmin
γ
⎛
T ⎞ 2( γ− 1)
Proved.
= ⎜ ηT ηC max ⎟
Tmin ⎠
⎝
∴
rP
⇒
If T3 = 1073 K, T1 = 300K, η1 = 0.8, η7 = 0.8, γ = 1.4 then
1.4
⎞ 2(1.4 – 1)
= 4.26
( rp )opt. = ⎛⎜⎝ 0.8 × 0.8 × 1073
⎟
300 ⎠
γ− 1
(rp )optγ = x = 1.513
⎡
1 ⎞ T (x − 1) ⎤
⎛
∴ Wmax = Cp ⎢ηT T3 ⎜1 − ⎟ − 1
⎥
⎝
x⎠
ηc
⎣
⎦
⎡
⎤
1 ⎞ 300
⎛
= 1.005 ⎢0.8 × 1073 ⎜1 −
(1.513 − 1) ⎥ kJ/kg
⎟−
1.513 ⎠ 0.08
⎝
⎣
⎦
= 99.18 kJ/kg
x − 1⎤
⎡
Heat input Q1 = h3 – h2 = Cp (T3 – T2)
T2 = T1 ⎢1 +
ηc ⎥⎦
⎣
= 1.005 (1073 – 492.4)
= 583.5 kJ/kg
= 492.4 K
Page 241 of 265
Gas Power Cycles
Chapter 13
∴η=
99.18
× 100% = 17%
583.5
Q13.29
A gas turbine plant draws in air at 1.013 bar, 10°C and has a pressure
ratio of 5.5. The maximum temperature in the cycle is limited to 750°C.
Compression is conducted in an uncooled rotary compressor having an
isentropic efficiency of 82%, and expansion takes place in a turbine with
an isentropic efficiency of 85%. A heat exchanger with an efficiency of
70% is fitted between the compressor outlet and combustion chamber.
For an air flow of 40 kg/s, find (a) the overall cycle efficiency, (b) the
turbine output, and (c) the air-fuel ratio if the calorific value of the fuel
used is 45.22 MJ/kg.
(Ans. (a) 30.4%, (b) 4272 kW, (c) 115)
Solution:
p1 = 101.3 kPa
T1 = 283 K
p2
= 5.5 kPa
p1
T4 = 750°C = 1023 K
γ−1
T
⎛p ⎞ γ
∴ 2s = ⎜ 2 ⎟ ⇒ T2s = 460.6 K
T1
⎝ p1 ⎠
T −T
T −T
ηc = 2s 1 ∴ T2 = T1 + 2s 1
T2 − T1
ηc
= 499.6K
T
⎛p ⎞
∴ 5s = ⎜ 5 ⎟
T4
⎝ p4 ⎠
γ− 1
γ
⎛p ⎞
= ⎜ 1⎟
⎝ p2 ⎠
γ− 1
γ
⎛ 1 ⎞
= ⎜
⎟
⎝ 5.5 ⎠
γ− 1
γ
p2
4 1023 K
+m
(1
g
)k
3
T
2s
K
6
0.
46 1 kg
(1 + m) kg
2
499.6 K
5
p1
687.7 K
5s
628.6 K
283 K 1
S
∴
⎛ 1 ⎞
T5s = T4 × ⎜
⎟
⎝ 5.5 ⎠
ηT =
1.4 − 1
1.4
= 628.6K
T4 − T5
∴ T4 – T5 = ηT (T4 – T5s) = 335.3 K
T4 − T5s
Page 242 of 265
Gas Power Cycles
Chapter 13
∴
T5 = 687.K
Maximum possible heat from heat exchanger = Cp (T5 – T2)
∴ Actual heat from = 0.7Cp (T5 – T2) = 132.33 kJ/kg of air
∴ Cp (T3 – T2) = (1 + m) 132.33 and
CpT3 = 132.33 +132.33 m + CpT2 = 634.43 +132.33 m
Heat addition (Q1) = Cp (T4 – T3) = CpT4 – CpT3
= 393.7 –132.33m = m × 45.22×103
∴
m = 8.68 × 10–3 kJ/kg of air
∴
Q1 = 392.6 kJ/kg of air
WT = (1 + m) (h4 –h5) = (1 + m) Cp (T4 – T5)
= 1.00868 × 1.005 × (1023 – 687.7) kJ/kg of air 340 kJ/kg
Wc = (h2 – h1) = Cp (T2 – T1) = 1.005 × (499.6 – 283)
= 217.7 kJ/kg of air
∴
Wnet = WT - Wc = 122.32 kJ/kg
122.32
× 100% = 31.16%
(a)
η=
392.6
(b) Turbine output = (WT) = 122.32 kJ/kg of air
= 4893 kW
1 kg air
= 115.2 kg of air/kg of fuel
0.00868 kg of fuel
A gas turbine for use as an automotive engine is shown in Fig. 13.43. In
the first turbine, the gas expands to just a low enough pressure p5 , for
(c) Air fuel ratio =
Q13.30
the turbine to drive the compressor. The gas is then expanded through a
second turbine connected to the drive wheels. Consider air as the
working fluid, and assume that all processes are ideal. Determine (a)
pressure p5 (b) the net work per kg and mass flow rate, (c) temperature
T3 and cycle thermal efficiency, and (d) the T − S diagram for the cycle.
Page 243 of 265
Gas Power Cycles
Chapter 13
Solution :
Try please.
Q13.31
Repeat Problem 13.30 assuming that the compressor has an efficiency of
80%, both the turbines have efficiencies of 85%, and the regenerator has
an efficiency of 72%.
Try please.
Solution:
Q13.32
An ideal air cycle consists of isentropic compression, constant volume
heat transfer, isothermal expansion to the original pressure, and
constant pressure heat transfer to the original temperature. Deduce an
expression for the cycle efficiency in terms of volumetric compression
ratio rk , and isothermal expansion ratio, rk In such a cycle, the pressure
and temperature at the start of compression are 1 bar and 40°C, the
compression ratio is 8, and the maximum pressure is 100 bar. Determine
the cycle efficiency and the m.e.p.
(Ans. 51.5%, 3.45 bar)
Solution:
V=C
3
Q1′
2
3
p
T=C
T
Q1′
2
T=C
4
Q2
S=C
r
pV = C
1
1
Q2 4
S
V
∴
p=C
Q1 ′′
Q1 ′′
Compression ratio, rc =
V1
V2
V4
V
= 4
V3
V2
Heat addition Q1 = Q1′ + Q1′′
= constant volume heat addition
(Q1′ + constant temperature heat addition Q1′′)
Heat rejection, Q2 = Cp (T4 – T1)
Expansion ratio, re =
∴ T2 – T1 . rcγ − 1
T3 = T4
p1 v3
p v
∴
= 4 4
T3
T4
∴ p3 = p1 . re
γ −1
γ
γ −1
⎛v ⎞
= rcγ − 1
= ⎜ 1⎟
v
⎝ 2⎠
v
and v2 = 1 and p 2 = p1 rcγ
rc
T
⎛p ⎞
Hence 2 = ⎜ 2 ⎟
T1
⎝ p1 ⎠
∴
p
p
v4
= 3 = 3 = re
p4
p1
v3
Page 244 of 265
Gas Power Cycles
Chapter 13
∴
∴
p
p2
= 3
T2
T3
or
η= 1−
p3
r
r
× T2 = eγ × T1 . rcγ - 1 = T1 e = T4
p2
rc
rc
Cp (T4 − T1 )
T3 =
Q2
= 1−
Cv (T3 − T2 ) + RT3 In re
Q1
r
⎛
⎞
Cp ⎜ T1 . e − T1 ⎟
rc
⎝
⎠
= 1−
re
r
⎛
⎞
Cv ⎜ T1 . − T1 rcγ − 1 ⎟ + R .T1 e In re
rc
rc
⎝
⎠
⎛r
⎞
γ ⎜ e − 1⎟
r
⎝ c
⎠
= 1−
r
re
⎛ e
γ −1 ⎞
⎜ r − rc ⎟ + ( γ − 1) r In re
c
⎝ c
⎠
γ(re − rc )
= 1−
γ
(re − rc ) + ( γ − 1) re l n re
∴
η= 1−
γ[re − rc ]
(re − r ) + ( γ − 1) re l n re
γ
c
Given p1 = 1 bar = 100 kPa
T1 = 40°C = 313 K
rc =8 and p3 = 100 bar = 10000 kPa
p3
= 100
p1
1.4 (100 − 8)
∴η= 1−
1.4
(100 − 8 ) + (1.4 – 1 × In 100
128.8
= 1−
265.83
= 0.51548 = 51.548 %
∴ p3 = p1 . re ∴ re =
⇒ T3 = T1 ×
re
313 × 100
=
= 3912.5 K
8
rc
T2 = T1 × rcγ − 1 = 719 K
∴ Heat addition, Q = Cv ( T3 –T2) + R T3 In re
= 0.718 (3912.5 – 719) + 0.287 × 3912.5 × ln 100
= 7464 kJ/kg
∴ Work, W = Q η = 3847.5 kJ/kg
∴ p m (V4 – V2) = W
∴ v 4 = 100 v 2
∴ p m (100 –1) v 2 = W
∴ pm (99) ×
v1
=W
8
Page 245 of 265
v2 =
v1
rc
Gas Power Cycles
Chapter 13
8W
= 346.1 kPa
99 × v1
= 3.461 bar
( v 4 – V3) = 40 58
∴ pm =
∴ pm
∴ pm =
Q13.37
v1 =
RT1
= 0.89831 kJ/kg
p1
4058
= 365 bar
v4
v4 −
100
Show that the mean effective pressure, pm ' for the Otto cycle is
Given by
(p
3
pM =
⎛
1 ⎞
− p1 rkγ ⎜1 − γ-1 ⎟
⎝ rk ⎠
( γ − 1)( rk −1)
)
Where p3 = pmax ' p1 = pmin and rk is the compression ratio.
Solution:
Intake p1 , v1 , T1
γ -1
T
⎛p ⎞ γ
⎛v ⎞
∴ 2 = ⎜ 2⎟
= ⎜ 1⎟
T1
⎝ p1 ⎠
⎝ v2 ⎠
γ −1
∴ T2 = T1 . rc
γ −1
= rcγ − 1
3
p
γ
PV = C
Q1
2
pv = C
γ
pv = C
v2 =
V
v1
rc
p3
p
= 2
T3
T2
∴ T3 = T2 ×
p3
p
r γ − 1 × p3
T ×p
= T2 × 3 γ = T1 c
= 1 3
γ
rc p1
p2
p1 rc
p1 rc
γ −1
γ −1
T3
⎛v ⎞
⎛v ⎞
= ⎜ 4 ⎟ = ⎜ 1 ⎟ = rcγ− 1
T4
⎝ v2 ⎠
⎝ v3 ⎠
T
T1 p3
T p
= 1γ 3
∴ T4 = γ −3 1 =
γ -1
rc
rc p1 × rc
rc p1
W = Q1 – Q2
= Cv (T3 – T2) – Cv (T4 – T1)
pm (V1 – V2) =W
∴
∴ pm =
4
γ
p2 = p1 × rcγ
Cv [(T3 − T2 ) − (T4 − T1 )]
V1 − V2
Page 246 of 265
Q2
1
Gas Power Cycles
Chapter 13
T p
⎡T p
⎤
cv ⎢ 1 3 − T1 rcγ - 1 − 1γ 3 + T1 ⎥
rc p1
⎣ rc p1
⎦
=
v1
v1 −
rc
p
⎡
⎤
p3 − p1 rcγ − γ −3 1 + p1 rc ⎥
cV T1 ⎢
rc
⎢
⎥
=
V1 p1 ⎣⎢
(rc − 1)
⎦⎥
R
⎡
⎢ cV = γ − 1
⎢
⎢⎣∵ p1 V1 = RT1
RT1
=
V1 p1
p3
+ ( p3 – p1 rcγ )]
rcγ − 1
( γ − 1) (rc − 1)
[( p3 − p1 rcγ ) −
1 ⎞
⎛
( p3 − p1 rcγ ) ⎜1 − γ − 1 ⎟
rc
⎝
⎠ Proved
=
( γ − 1)(rc − 1)
Q13.38
A gas turbine plant operates on the Bray ton cycle using an optimum
pressure ratio for maximum net work output and a regenerator of 100%
effectiveness. Derive expressions for net work output per kg of air and
corresponding efficiency of the cycle in terms of the maximum and the
minimum temperatures.
If the maximum and minimum temperatures are 800°C and 30°C
respectively, compute the optimum value of pressure ratio, the
maximum net work output per kg and the corresponding cycle
efficiency.
2
T
(Ans. (Wnet )max = C p Tmax − Tmin
(ηcycle )max = 1 − Tmin , ( rp )opt = 9.14
max
(
)
(Wnet )max = 236.97 kJ/kg;ηcycle = 0.469 )
T1 = Tmin
Solution:
T4 = Tmax
∴
∴
T2
⎛p ⎞
= ⎜ 2⎟
T1
⎝ p1 ⎠
T2 = T1 x
γ− 1
γ
γ −1
γ− 1
= rp γ = x (say)
γ −1
T5 ⎛ p5 ⎞ γ
1
⎛p ⎞ γ
= ⎜ ⎟ = ⎜ 1⎟
=
x
T4 ⎝ p4 ⎠
⎝ p2 ⎠
T
∴
T5 = 4
x
For regeneration 100% effective number
Cp (T5 – T2) = Cp (T3 – T2)
T
∴
T3 = T5 = 4
x
WT = h4 – h5 = Cp (T4 – T5)
Page 247 of 265
Gas Power Cycles
Chapter 13
T ⎞
⎛
= Cp ⎜ T4 − 4 ⎟
⎝
x ⎠
p2
4
Q1
3
T
p1
2
5
Q2
1
S
And
1⎞
⎛
= Cp T4 ⎜1 − ⎟
⎝
x⎠
Wc = h2 – h1
= Cp (T2 – T1)
= Cp T1 (x – 1)
⎡ ⎛
⎤
1⎞
Wnet = WT – WC = Cp ⎢T4 ⎜1 − ⎟ − T1 (x − 1) ⎥
x⎠
⎣ ⎝
⎦
For Maximum Net work done
∂ Wnet
1
= 0 ∴ T4 × 2 − T1 = 0
∂x
x
T
T
∴
x2 = 4 = max
Tmin
T1
∴
∴
x=
Tmax
Tmin
γ
⎛ T ⎞ 2( γ − 1)
Heat addition ∴ ( rp ) opt. = ⎜ max ⎟
⎝ Tmin ⎠
T ⎞
⎛
Q1 = h4 – h3 = Cp (T4 – T3) = Cp⎜ T4 − 4 ⎟
⎝
x ⎠
1⎞
⎛
= Cp T4 ⎜1 − ⎟
⎝
x⎠
⎡
T1 ⎤
= Cp T4 ⎢1 −
⎥
T4 ⎥⎦
⎢⎣
1⎞
⎛
T4 ⎜1 − ⎟ − T1 (x − 1)
Wnet
x⎠
⎝
∴
η opt. =
=
1⎞
Q1
⎛
T4 ⎜1 − ⎟
x⎠
⎝
= 1−
T
T4
T1
×x = 1− 1 ×
= 1−
T4
T4
T1
Page 248 of 265
Tmin
Tmax
Gas Power Cycles
Chapter 13
Wopt. = Cp [T4 − T1T4 − T1T4 + T1 ]
= Cp [ T4 − T1 ]2 = Cp [ Tmax − Tmin ]2
If Tmax = 800°C = 1073 K;
∴
⎛ 1073 ⎞
rp,opt = ⎜
⎟
⎝ 303 ⎠
η opt. = 1 −
Tmin = 30°C = 303K
1.4
2(1.4–1)
= 9.14
Tmin
= 46.9%
Tmax
Wopt. = 1.005 ( 1073 − 303)2 = 236.8 kJ/kg
Q13.40
Show that for the Sterling cycle with all the processes occurring
reversibly but where the heat rejected is not used for regenerative
heating, the efficiency is giver: by
⎛ T1
⎞
− 1 ⎟ + (γ − 1) ln r
⎜
T
⎠
η =1 − ⎝ 2
⎛ T1
⎞
T1
⎜ − 1 ⎟ + (γ − 1) ln r
T2
⎝ T2
⎠
Where r is the compression ratio and T1 / T2 the maximum to minimum
temperature ratio.
Determine the efficiency of this cycle using hydrogen (R = 4.307 kJ/kg K,
c p =. 14.50 kJ/kg K) with a pressure and temperature prior to isothermal.
Compression of 1 bar and 300 K respectively, a maximum pressure of2.55
MPa and heat supplied during the constant volume heating of 9300
kJ/kg. If the heat rejected during the constant volume cooling can be
utilized to provide the constant volume heating, what will be the cycle
efficiency? Without altering the temperature ratio, can the efficiency be
further improved in the cycle?
Solution:
Minimum temperature
(T2) = Tmin
4
Maximum temperature
(T1) = Tmax
∴
∴
p Q
1
T=C
1
3
Compression ratio
v
v
( rc ) = 2 = 1
v3
v4
Q2
T=C
T1 – T4
and
T3 = T2
v
WT = RT1 ln 1 = RT1 ln rc
v4
∴
Q2
⎛v ⎞
WC = RT2 ln ⎜ 2 ⎟ = RT2 ln rc
⎝ v3 ⎠
Wnet =R ln ( rc ) × [T1 – T2]
Page 249 of 265
2
Gas Power Cycles
Chapter 13
Constant volume Heat addition (Q1) = Cv (T1 – T2)
R
(T1 − T2 )
=
γ −1
Constant temperature heat addition Q2 = RT2 ln rc
(T − T2 ) ⎤
⎡
∴ Total heat addition Q = Q1 + Q2 = R ⎢T1 ln rc 1
( γ − 1) ⎥⎦
⎣
( γ − 1) ln rc (T1 − T2 )
W
ln rc [T1 − T2 ]
=
−1 +1
η = net =
T2 − T1 ⎤
( γ − 1) T1 ln rc − (T2 − T1 )
Q
⎡
⎢T1 ln rc − r − 1 ⎥
⎣
⎦
( γ − 1) ln rc (T1 − T2 ) ⎤
⎡
= 1 − ⎢1 −
( γ − 1) ln rc − (T2 − T1 ) ⎥⎦
⎣
( γ − 1) T1 ln rc − (T2 − T1 ) − ( γ − 1) ln rc T1 + ( γ − 1) T2 ln rc
= 1−
( γ − 1) ln rc − (T2 − T1 )
= 1−
(T1 − T2 ) + ( γ − 1) T2 ln rc
(T1 − T2 ) + ( γ − 1) T1 ln rc
⎛ T1
⎞
⎜ T − 1 ⎟ + ( γ − 1) ln rc
⎠
= 1− ⎝ 2
Proved
T1
⎛ T1
⎞
⎜ T − 1 ⎟ + ( γ − 1) T ln rc
2
⎝ 2
⎠
Q13.41
Helium is used as the working fluid in an ideal Brayton cycle. Gas enters
the compressor at 27 °C and 20 bar and is discharged at 60 bar. The gas is
heated to l000 °C before entering the turbine. The cooler returns the hot
turbine exhaust to the temperature of the compressor inlet. Determine:
(a) the temperatures at the end of compression and expansion, (b) the
heat supplied, the heat rejected and the net work per kg of He, and (c) the
cycle efficiency and the heat rate. Take c p = 5.1926 kJ/kg K.
(Ans. (a) 4 65.5, 820.2 K, (b) 4192.5, 2701.2, 1491.3 kJ/kg,
(c) 0.3557, 10,121kJ/kWh)
T2
⎛p ⎞
= ⎜ 2⎟
T1
⎝ p1 ⎠
Solution:
∴
γ −1
γ
=
60
20
Cp = 5.1926, R = 2.0786
c v = c p – R = 3.114
γ =
cp
cv
=
5.1926
= 1.6675
3.114
∴
γ −1
= 0.4
γ
Page 250 of 265
Gas Power Cycles
Chapter 13
p2
3
1273 K
Q1
T
2
465.7 K
p1
4
820 K
Q2
1 300 K
S
∴
⎛ 60 ⎞
T2 = T1 × ⎜ ⎟
⎝ 20 ⎠
γ −1
γ
= 465.7 K
∴
T
⎛p ⎞
∴ 4 = ⎜ 4⎟
T3
⎝ p3 ⎠
γ −1
γ
⎛ 20 ⎞
= ⎜ ⎟
⎝ 60 ⎠
γ −1
γ
∴ T4 = T3 ×
1
γ − 1 = 820 K
3 γ
(a)
End of compressor temperature T2 = 465.7K
End of expansion temperature T4 = 820K
(b)
Heat supplied (Q1) = h3 – h2 = CP (T3 –T2) = 4192 kJ/kg
Heat rejected (Q2) = h4 – h1 = CP (T4 –T1) = 2700 kJ/kg
Work,
W = Q1 – Q2 = 1492 kJ/kg
W
1492
× 100% = 35.6%
=
Q1
4192
3600
3600
Heat rate =
= 10112 kJ/kWh
=
0.356
η
(c)
Q13.42
Solution:
η=
An air standard cycle for a gas turbine jet propulsion unit, the pressure
and temperature entering the compressor are 100 kPa and 290 K,
respectively. The pressure ratio across the compressor is 6 to 1 and the
temperature at the turbine inlet is 1400 K. On leaving the turbine the air
enters the nozzle and expands to 100 kPa. Assuming that the efficiency
of the compressor and turbine are both 85% and that the nozzle
efficiency is 95%, determine the pressure at the nozzle inlet and the
velocity of the air leaving the nozzle.
(Ans. 285 kPa, 760 m / s)
p2
=6
∴ p2 = 600 kPa
p1
Page 251 of 265
Gas Power Cycles
Chapter 13
3
p2
pi
5
T
2s
2
4s
p1
6
1
290 K, 100 kPa
S
γ −1
1.4 − 1
T2s
⎛p ⎞ γ
= ⎜ 2⎟
= 6 1.4
T1
⎝ p1 ⎠
T2s = 483.9 K
T2s − T1
T2 − T1
T − T1
= 228 K
∴ T2 – T1 = 2s
ηc
T2 = 518 K
T3 = 1400 K
WC = CP (T2 – T1) = 1.005 (518 – 290) = 229.14 kJ/kg
ηC =
γ −1
T4 s
⎛p ⎞ γ
= ⎜ i⎟
T3
⎝ p2 ⎠
W
∴
WT = C = 269.9 kJ/kg = CP (T3 – T4s)
ηT
∴ T3 – T4s = 268.24
∴
T4s = 1131.8 K
1.4
∴
p
⎛ 1131.8 ⎞1.4 − 1
= i
⎜
⎟
p2
⎝ 1400 ⎠
1.4
∴
pi = p2
⎛ 1131.8 ⎞1.4 − 1
× ⎜
= 285 kPa
⎟
⎝ 1400 ⎠
Δh = h5 – h6 = CP (T5 – T6)
T3 − T5
= ηT
T3 − T4 s
T5
⎛p ⎞
= ⎜ 5⎟
T6
⎝ p6 ⎠
∴
∴ T3 – T5 = 227.97
γ −1
γ
⎛ 285 ⎞
= ⎜
⎟
⎝ 100 ⎠
1.4 − 1
1.4
∴ T5 = 1172 K
⇒ T6 = T5 = 868.9 K
Δh = CP (1172 – 868.9) = 304.6 kJ/kg
Page 252 of 265
Gas Power Cycles
Chapter 13
∴
Q13.43
V=
2000 × η × Δ h =
2000 × 0.95 × 304.6 m/s = 760.8 m/s
A stationary gas turbine power plant operates on the Brayton cycle and
delivers 20 MW to an electric generator. The maximum temperature is
1200 K and the minimum temperature is 290 K. The minimum pressure is
95 kPa and the maximum pressure is 380 kPa. If the isentropic
efficiencies of the turbine and compressor are 0.85 and 0.80 respectively,
find (a) the mass flow rate of air to the compressor, (b) the volume flow
rate of air to the compressor, (c) the fraction of the turbine work output
needed to drive the compressor, (d) the cycle efficiency.
If a regenerator of 75% effectiveness is added to the plant, what would be
the changes in the cycle efficiency and the net work output?
(Ans. (a) 126.37 kg/s, (b) 110.71 m3 /s, (c) 0.528,
(d) 0.2146, Δη = 0.148 ΔWnet = 0)
T2
⎛p ⎞
= ⎜ 2⎟
T1
⎝ p1 ⎠
Solution:
T4 ⎛ p4 ⎞
=
T3 ⎜⎝ p3 ⎟⎠
γ− 1
γ
γ −1
γ
∴ T2 = 431K
⎛p ⎞
=⎜ 1 ⎟
⎝ p2 ⎠
γ −1
γ
; T4 = 807.5 K
∴
Wnet = (h3 – h4) – (h2 – h1)
= CP [(T3 – T4) – (T2 – T1)]
= 252.76 kJ/kg
20000
•
= 79.13 kg/s
∴ Mass flow rate (m) =
252.76
3
p2
380 kPa
1200 K
431 K
T
p1
2
4
807.5 K
1 95 kPa, 290 K
S
•
(a) Turbine output (WT) = m cP (T3 – T4) = 31.234 MW
(b) η =
WC
T − T1
= 0.3592
= 2
T3 − T4
WT
Page 253 of 265
Gas Power Cycles
Chapter 13
•
(c) (m) = 79.13 kg/s
(d) v1 =
RT1
= 0.8761 m3/kg
p1
•
•
∴ V = mv1 = 69.33 m3/s
Page 254 of 265
Refrigeration Cycles
Chapter 14
14. Refrigeration Cycles
Some Important Notes
Heat Engine, Heat Pump
Heat engines, Refrigerators, Heat pumps:
•
A heat engine may be defined as a device that operates in a thermodynamic cycle and does
a certain amount of net positive work through the transfer of heat from a high
temperature body to a low temperature body. A steam power plant is an example of
a heat engine.
•
A refrigerator may be defined as a device that operates in a thermodynamic cycle and
transfers a certain amount of heat from a body at a lower temperature to a body at a
higher temperature by consuming certain amount of external work. Domestic
refrigerators and room air conditioners are the examples. In a refrigerator, the required
output is the heat extracted from the low temperature body.
•
A heat pump is similar to a refrigerator, however, here the required output is the heat
rejected to the high temperature body.
Fig. (a) Heat Engine (b) Refrigeration and heat pump cycles
Page 255 of 265
Refrigeration Cycles
Chapter 14
Fig. Comparison of heat engine, heat pump and refrigerating machine
QH
QH
TH
=
=
Wcycle QH − QC
TH − TC
COPCarnot,HP =
COPCarnot,R =
Where
Wcycle =
QH
=
QC
=
TH
TC
=
=
QC
QC
TC
=
=
Wcycle QH − QC
TH − TC
work input to the reversible heat pump and refrigerator
heat transferred between the system and the hot reservoir
heat transferred between the system and cold reservoir
temperature of the hot reservoir.
temperature of the cold reservoir.
Page 256 of 265
Refrigeration Cycles
Chapter 14
Question and Solution (P K Nag)
Q14.1
Solution:
Q14.2
A refrigerator using R–134a operates on an ideal vapour compression
cycle between 0.12 and 0.7 MPa. The mass flow of refrigerant is 0.05 kg/s.
Determine
(a) The rate of heat removal from the refrigerated space
(b) The power input to the compressor
(c) The heat rejection to the environment
(d) The COP
(Ans. (a) 7.35 kW, (b) 1.85 kW, (c) 9.20 kW, (d) 3.97)
Try please.
A Refrigerant-12 vapour compression cycle has a refrigeration load of 3
tonnes. The evaporator and condenser temperatures are – 20°C and 40°C
respectively. Find
(a) The refrigerant flow rate in kg/s
(b) The volume flow rate handled by the compressor in m3/s
(c) The work input to the compressor in kW
(d) The heat rejected in the condenser in kW
(e) The isentropic discharge temperature.
If there is 5o C of superheating of vapour before it enters the
compressor, and 5o C sub cooling of liquid before it flows through the
expansion valve, determine the above quantities.
Solution: As 50°C temperature difference in evaporate so evaporate temperature = – 20°C and
Condenser temperature is 30°C.
∴ p1 = 1.589 bar
4 3
p2 = 7.450 bar
2
h7 = 178.7 kJ/kg, h3 = 64.6 kJ/kg
5
p
h1 = 178.7 +
(190.8 – 178.7)
20
1
7
5 6
Δh = 3.025 kJ/kg
5
s1 = 0.7088 +
(0.7546 – 0.7088)
h
20
= 0.7203 kJ/kg– K
[Data from CP Arora]
∴ h3 – h4 = Δh = h1 – h7 = 3.025
∴ h4 = h3 – Δh = 61.6 kJ/kg i.e.
25°C hg = 59.7
30°C hg = 64.6 → 0.98/vc
∴ Degree of sub cooling = 3.06°C
0.7203 – 0.6854
× 20 = 15°C
(a) Degree of super heat is discharge =
0.7321 – 0.6854
∴
Discharge temperature = 15 + 30 = 45° C
15
(214.3 − 199.6) = 210.63 kJ/kg
∴ h2 = 199.6 +
20
∴ Compressor work (W) = h2 – h1 = 210.63 – 181.73 = 28.9 kJ/kg
Refrigerating effect (Q0) = h7 – h5 = h7 – h4 = (178.7 – 61.6) kJ/kg = 117.1 kJ/kg
Page 257 of 265
Refrigeration Cycles
Chapter 14
∴
(b)
COP =
Qo
117.1
=
= 4.052
W
28.9
•
v1 = 0.108 m3 /kg
•
V1 = mv1 = 0.014361 m3/s
•
π D2
N
×L×
× n × ηvol = V1
4
60
L
= 1.2
D
L = 1.2 D
π × D2
900
× 1.2 D ×
× 1 × 0.95 = 0.014361
4
60
∴ D = 0.1023 m = 10.23 cm
L = 0.1227 m = 12.27 cm
Q14.4
A vapour compression refrigeration system uses R-12 and operates
between pressure limits of 0.745 and 0.15 MPa. The vapour entering the
compressor has a temperature of – 10°C and the liquid leaving the
condenser is at 28°C. A refrigerating load of 2 kW is required. Determine
the COP and the swept volume of the compressor if it has a volumetric
efficiency of 76% and runs at 600 rpm.
(Ans. 4.15, 243 cm3)
Solution:
p1 = 150 kPa: Constant saturated temperature (– 20°C)
p2 = 745 kPa: Constant saturated temperature (30°C)
ding
o
c
b
u
2°C s
4 3
2
p
5 6
7
1
10°C superheated
h7 = 178.7 kJ/kg
h3 = 64.6 kJ/kg
h4 = h4-5 = 59.7 +
h
3
(64.6 – 59.7) = 62.64 kJ/kg = h5
5
10
(190.8 – h7 ) = 184.8 kJ/kg
20
10
(0.7546 – 0.7088) = 0.7317 kJ/kg-K
= 0.7088 +
20
h1 = h7 +
s1
Page 258 of 265
Refrigeration Cycles
Chapter 14
⎛ 0.7317 – 0.6854 ⎞
h2 = 199.6 + ⎜
⎟ (214.3 – 199.6) = 214.2 kJ/kg
⎝ 0.7321 – 0.6854 ⎠
∴ Compressor work (W) = h2 – h1 = 29. 374 kJ/kg
Refrigeration effect = (h1 – h5) = (184.8 – 62.64) = 122.16 kJ/kg
122.16
= 4.16
∴
COP =
29.374
v1 = 0.1166m3 /kg
•
Mass flow ratio m × 122.16 = 2
•
∴ m = 0.016372 kg/s
•
•
∴ V1 = mv1 = 1.90897 × 103 m3/s = Vs × 0.76 ×
600
60
∴ Vs = 251.2 cm3
Q14.6
Solution:
A R-12 vapour compression refrigeration system is operating at a
condenser pressure of 9.6 bar and an evaporator pressure of 2.19 bar.
Its refrigeration capacity is 15 tonnes. The values of enthalpy at the
inlet and outlet of the evaporator are 64.6 and 195.7 kJ/kg. The specific
volume at inlet to the reciprocating compressor is 0.082 m3/kg. The
index of compression for the compressor is 1.13
Determine:
(a) The power input in kW required for the compressor
(b) The COP. Take 1 tonnes of refrigeration as equivalent to heat
removal at the rate of 3.517 kW.
(Ans. (a) 11.57 kW, (b) 4.56)
T1 = – 10°C
T3 = 40°C
h4 = 646 kJ/kg
h1 = 1057 kJ/kg
n = 1.13
v1 = 0.082 m3 / kg
3
2
2′
p
4
1
1′
h
Refrigeration effect (195.7 – 64.6) kJ/kg = 131.1 kJ/kg
•
•
∴ m = 0.4024 kg/s
m Qo = 15 × 3.517
1
1
v2
⎛ p ⎞n
⎛ 2.19 ⎞1.3
= ⎜ 1⎟ = ⎜
⎟ = v 2 = 0.022173 m3/kg
v1
p
9.6
⎝
⎠
⎝ 2⎠
n
∴ WC =
( p1V1 − p2V2 ) = 28.93 kJ/kg
n −1
(a) Wcompressor = 11.64 KW
Page 259 of 265
Refrigeration Cycles
Chapter 14
(b)
Q14.12
Solution :
COP =
15 × 3.517
= 4532
11.64
Determine the ideal COP of an absorption refrigerating system in
which the heating, cooling, and refrigeration take place at 197°C, 17°C,
and –3°C respectively.
(Ans. 5.16)
Th
Desired effort
Qh
∴ COP =
input
W
HE
Refregerating effect
(Qh –W)
=
heat input
Ta
(Qo+ W)
Qo
=
R1
W
Qh
=
Qo W
×
W Qh
Qo
To
= (COP) R × η H.E.
For ideal process
(COP)R =
And
To
Ta − To
T ⎞
⎛
ηH.E = ηCarnot = ⎜1 − a ⎟
Th ⎠
⎝
To
T ⎞
⎛
× ⎜1 − a ⎟
Ta − To ⎝
Th ⎠
T [T − Ta ]
= o × h
Th [Ta − To ]
∴ (COP) ideal =
Given To = 270 K, Ta = 290 K, Th = 470 K
∴ (COP) ideal =
Q14.22
Solution:
270 [470 − 290]
×
= 5.17
470 [290 − 270]
Derive an expression for the COP of an ideal gas refrigeration cycle with
a regenerative heat exchanger. Express the result in terms of the
minimum gas temperature during heat rejection (Th) maximum gas
temperature during heat absorption (T1) and pressure ratio for the cycle
T1
⎛
⎞
( p2 p1 ) .
⎜ Ans. COP = T r ( γ −1) / γ − T ⎟
h p
1 ⎠
⎝
T
⎛p ⎞
∴ 2 = ⎜ 2⎟
T1
⎝ p1 ⎠
γ −1
γ
γ−1
γ
= rP
γ −1
γ
∴ T2 = T1 rP
Page 260 of 265
Refrigeration Cycles
Chapter 14
T4
⎛p ⎞
= ⎜ 4⎟
T5
⎝ p5 ⎠
∴
γ −1
γ
γ −1
γ −1
γ
⎛p ⎞ γ
=⎜ 2 ⎟
= rP
⎝ p1 ⎠
T
T
T5 = γ 4− 1 = γ n− 1
rP γ
rP γ
For Regeneration ideal
CP (T3 – T4) = p (T1 – T6)
∴ T3 – Th = T1 – T6
∴
Work input (W) = (h2 – h1) – (h4 – h5)
= CP [(T2 – T1) – (Th – Ts)
Heat rejection (Q1) = Q2 + W = CP (T2 – T3)
Heat absorption (Q2) = CP (T6 – T5)
p2
2
Q1
T
4
WC
Th 3
6
WE
1
QX
Q2
5 QX
T1
p1
S
∴
COP =
T6 − T5
Q2
1
=
=
Q1 − Q2
(T2 − T3 ) − (T6 − T5 ) T2 − T3
−1
T6 − T5
γ− 1
γ −1
T2 – T3 = T1 rP γ − T1 = T1 (rP γ − 1)
γ −1
T6 – T5 = Th −
Th
γ −1
rP γ
Q14.23
= Th
(rP γ − 1)
γ −1
rP γ
=
Th
γ −1
T1 rP γ − Th
or COP =
T
Th r
1
( γ −1)/ γ
p
− T1
Large quantities of electrical power can be transmitted with relatively
little loss when the transmission cable is cooled to a superconducting
temperature. A regenerated gas refrigeration cycle operating with
helium is used to maintain an electrical cable at 15 K. If the pressure
ratio is 10 and heat is rejected directly to the atmosphere at 300 K,
determine the COP and the performance ratio with respect to the
Carnot cycle.
(Ans. 0.02, 0.38)
Page 261 of 265
Refrigeration Cycles
Chapter 14
T2
⎛p ⎞
= ⎜ 2⎟
T1
⎝ p1 ⎠
Solution:
∴
γ− 1
γ
= 10
γ −1
γ
1.6667.1
T2 = 300 × 10 1.6667 = 754 K
T5
⎛p ⎞
= ⎜ 5⎟
T4
⎝ p4 ⎠
γ −1
γ
⎛ p⎞
= ⎜ ⎟
⎝ p2 ⎠
γ −1
γ
⎛1 ⎞
= ⎜ ⎟
⎝ 10 ⎠
0.4
2
Q1
T
300 K
15 K
5
3
1
4
6
Q2
S
∴
T5 = 5.9716 K
Refrigerating effect (Q2) = CP (T6 – T5) = 9.0284 CP
Work input (W) =CP [(T2 – T1) – (T4 – T5)] = 444. 97 CP
∴
COP =
And (COP) carnet =
0.0284 CP
= 0.0203
444.97 CP
T6
15
=
= 0.05263
T6 − T5
300 − 15
COP actual
0.0203
= 0. 3857
=
0.05263
COPcarnot
Q14.25
A heat pump installation is proposed for a home heating unit with an
output rated at 30 kW. The evaporator temperature is 10°C and the
condenser pressure is 0.5 bar. Using an ideal vapour compression cycle,
estimate the power required to drive the compressor if steam/water
mixture is used as the working fluid, the COP and the mass flow rate of
the fluid. Assume saturated vapour at compressor inlet and saturated
liquid at condenser outlet.
(Ans. 8.0 kW, 3.77, 0.001012 kg/s)
Page 262 of 265
Refrigeration Cycles
Chapter 14
h1 = 2519.8 kJ/kg
s1 = 8.9008 kJ/kg-K
v1 = 106.38 m3/kg
Solution:
3
81.33°C
hf = 340. 5 kJ/kg
2
50 kPa
0.5 bar
p
1
4
10°C
1.2266 kPa
h
400°C 50 kPa s = 8.88642,
500°C 50 kPa s = 9.1546,
h = 3278.9
h = 3488.7
⎛ 8.9008 − 8.8642 ⎞
h2 = (3488.7 – 3278.9) × ⎜
⎟ + 3278.9 = 3305.3 kJ/kg
⎝ 9.1546 − 8.8642 ⎠
Compressor work (W) = h2 – h1 = (3305.3 – 2519.8)
= 785.5 kJ/kg
∴
∴
Heating (Q) = h2 – hf3 = (3305.3 – 340.5) kJ/kg = 2964.8 kJ/kg
•
m × Q = 30
30
= 0.0101187 kg/s
2964.8
2964.8
COP =
= 3.77
785.5
•
= m =
∴
•
Compressor power = mW = 7.95 KW
Q14.26
A 100 tonne low temperature R-12 system is to operate on a 2-stage
vapour compression refrigeration cycle with a flash chamber, with the
refrigerant evaporating at – 40°C, an intermediate pressure of 2.1912 bar,
and condensation at 30°C. Saturated vapour enters both the compressors
and saturated liquid enters each expansion valve. Consider both stages
of compression to be isentropic. Determine:
(a) The flow rate of refrigerant handled by each compressor
(b) The total power required to drive the compressor
(c) The piston displacement of each compressor, if the clearance is 2.5%
for each machine
(d) The COP of the system
(e) What would have been the refrigerant flow rate, the total work of
compression, the piston displacement in each compressor and the
compressor and the COP, if the compression had occurred in a
single stage? .
(Ans. (a) 2.464, 3.387 kg/s, (b) 123 kW, (c) 0.6274, 0.314 m3/s, (d) 2.86,
(e) 3.349 kg/s, 144.54 kW, 1.0236 m3/s, 2.433)
Page 263 of 265
Refrigeration Cycles
Chapter 14
h1 = 169 kJ/kg
h3 = 183.2 kJ/kg
h5 = 64.6 kJ/kg = h6
h7 = h8 = 26.9 kJ/kg
Solution:
5
4
30°C.
m1
7
2.1912 bar p
i
6 –10° C 3
.
m2
p
8
9
–90°C
1
7.45 bar
p
2 2
p1 = 0.6417 bar
h
S1 = S2 = 0.7274 kJ/kg – K
S3 = S4 = 0.7020 kJ/kg – K
∴
From P.H chart of R12
h2 = 190 kJ/kg
h4 = 206 kJ/kg
•
•
•
•
m2 h 2 + m1h5 = m2 h7 + m1h 3
∴
•
•
m2 (h 2 − h7 ) = m2 (h3 − h5 )
•
m1
∴
•
m2
•
=
m2 (h1 – h8) =
(a)
h 2 − h7
190 − 26.9
=
= 1.3752
h3 − h5
183.2 − 64.6
100 × 14000
3600
•
∴ m2 = 2.7367 kg/s
•
m1 = m2 × 1.3752 = 3.7635 kg/s
•
•
(b) Power of compressor (P) = m2 (h 2 − h1 ) + m1 (h 4 − h3 )
= 14328 kW
(d) COP =
(e)
Refrigeration efficiency
100 × 14000
= 2.7142
=
Compressor
3600 × 143.28
For single storage
From R12 chart ha′ = 2154 kJ/kg, hg = hs = 64.6 kJ/kg
100 × 14000
•
•
∴
m(h1 − h 9 ) =
⇒ m = 3.725 kg/s
3600
•
Compressor power (P) = m (h4′ – h1) = 3.725 × 46
= 171.35 kW
Page 264 of 265
Refrigeration Cycles
Chapter 14
100 × 14000
3600
COP =
= 2.27
171.35
Page 265 of 265